---
abstract: 'A $\mathbb{T}$-gain graph is a simple graph in which a unit complex number is assigned to each orientation of an edge, and its inverse is assigned to the opposite orientation. The associated adjacency matrix is defined canonically, and is called $\mathbb{T}$-gain adjacency matrix. Let $\mathbb{T}_{G} $ denote the collection of all $\mathbb{T}$-gain adjacency matrices on a graph $G$. In this article, we study the cospectrality of matrices in $\mathbb{T}_{G} $ and we establish equivalent conditions for a graph $G$ to be a tree in terms of the spectrum and the spectral radius of matrices in $\mathbb{T}_{G} $. We identify a class of connected graphs $\mathfrak{F^{''}}$ such that for each $G \in \mathfrak{F^{''}}$, the matrices in $\mathbb{T}_G$ have nonnegative real part up to diagonal unitary similarity. Then we establish bounds for the spectral radius of $\mathbb{T}$-gain adjacency matrices on $ G \in \mathfrak{F^{''}} $ in terms of their largest eigenvalues. Thereupon, we characterize $\mathbb{T}$-gain graphs for which the spectral radius of the associated $\mathbb{T}$-gain adjacency matrices equal to the largest vertex degree of the underlying graph. These bounds generalize results known for the spectral radius of Hermitian adjacency matrices of digraphs and provide an alternate proof of a result about the sharpness of the bound in terms of largest vertex degree established in \[Krystal Guo, Bojan Mohar. Hermitian adjacency matrix of digraphs and mixed graphs. J. Graph Theory 85 (2017), no. 1, 217-248.\].'
author:
- 'Aniruddha Samanta[^1]'
- 'M. Rajesh Kannan[^2]'
bibliography:
- 'raj-ani-ref1.bib'
title: On the spectrum of complex unit gain graph
---

[**AMS Subject Classification(2010):**]{} 05C50, 05C22, 15B57.

**Keywords.** Gain graph, Suitably oriented graph, Normal spanning tree, Distinguished edge property, Perron-Frobenius theory, Spectral radius.

Introduction
============

For a given group $\mathfrak{G}$, a $\mathfrak{G}$-gain graph is a graph $G$ with each orientation of an edge of $G$ is assigned an element $g \in \mathfrak{G}$ (called gain of the oriented edge) and whose inverse $g^{-1}$ is assigned to the opposite orientation of the edge. For more details about the notion of $\mathfrak{G}$-gain graphs, we refer to [@Zaslav-gain-balance; @zas4; @Zaslav; @Zaslav-biased-graph-II; @Zaslav-biased-graph-III; @Zaslav-biased-graph-IV]. Let $\mathbb{T}=\{z\in\mathbb{C}:|z|=1\} $ be the multiplicative group of unit complex numbers. In [@reff1], the notion of $\mathbb{T}$-gain graphs has been studied. If $G$ is a graph with vertex set $ V(G)=\{v_{1}, v_{2}, \dots, v_{n} \} $ and each orientation of its edges having some gain from $\mathbb{T}$, then the associated $\mathbb{T}$-gain adjacency matrix is an $n \times n$ matrix defined as follows: $(s,t)th$ entry of the matrix is the gain of the edge starts from the vertex $v_{s}$ and ends at the vertex $v_{t}$, and is $0$ if there is no edge between the vertices $v_{s}$ and $v_{t}$. Particular cases of the notion of adjacency matrix of $\mathbb{T}$-gain graphs were considered with different gains in the literature [@Bap; @Kat]. In [@Kat], the authors considered complex weighted graphs with the weights are in $\{\pm1,\pm i\}$. In [@Ger], the authors studied some of the properties of the characteristic polynomial of the $\mathbb{T}$-gain graphs. For some interesting spectral properties of $\mathbb{T}$-gain graphs, we refer to [@reff1; @Reff2016; @Ger; @Zaslav; @Zas2]. A *directed graph(or digraph)* $ X $ is an order pair $ (V(X), E(X)) $, where $ V(X)=\{ v_{1}, v_{2}, \dots,v_{n}\} $ is the vertex set and $ E(X) $ is the directed edge set. A directed edge from the vertex $ v_{s} $ to the vertex $ v_{t} $ is denoted by $ \overrightarrow{e_{st}} $. If $ \overrightarrow{e_{st}} \in E(X)$ and $  \overrightarrow{e_{ts}}\in E(X)$, then the pair $ \{v_{s},v_{t}\} $ is called a *digon* of $ X $. The underlying graph of $ X $ is a simple undirected graph obtain from $ X $ by replacing a directed edge by an undirected edge and denoted by $ \Gamma(X) $. The *Hermitian adjacency matrix* [@Bojan; @Lie] of a digraph $ X $ is denoted by $H(X)$ and is defined as follows: $$\mbox{$ (s,t){th}$ entry of }H(X)=h_{st}=\begin{cases}
    1& \text{if both } \mbox{$\overrightarrow{e_{st}}$ \text{and} $\overrightarrow{e_{ts}} \in E(X)$},\\
    i& \text{if  } \mbox{$\overrightarrow{e_{st}} \in E(X)$ \text{and} $\overrightarrow{e_{ts}} \notin E(X)$},\\
    -i& \text{if  } \mbox{$\overrightarrow{e_{st}} \notin E(X)$ \text{and} $\overrightarrow{e_{ts}} \in E(X)$},\\
    0&\text{otherwise.}\end{cases}$$

The Hermitian adjacency matrix can be thought of as the adjacency matrix of a $\mathbb{T}$-gain graph with the gains are from $ \{1, \pm i\}$.

Given a graph $G$, one can associate uncountably many $\mathbb{T}$-gain graphs with $G$. Let $\mathbb{T}_{G} $ denote the collection of all $\mathbb{T}$-gain adjacency matrices on a graph $G$. First main objective of this article is to study the relationship between the spectral properties of various $\mathbb{T}$-gain adjacency matrices associated with $G$. In section \[co-spec\], we partition $\mathbb{T}_{G} $ using the gains of some special types of cycles(fundamental cycles) of $G$, and show that any two matrices inside the same class are cospectral\[Theorem \[Theo3.2\]\]. Also we observe that, either the specturm or the spectral radius of $\mathbb{T}$-gain adjacency matrices of a graph $G$ is invariant under all $\mathbb{T}$-gains if and only if the $G$ is a tree\[Theorem \[spec-equi-tree\]\]. We identified a class of connected graphs such that for each graph $G$ in this class the cospectrality of matrices in $\mathbb{T}_G$ is determined by the gains of the fundamendal cycles.

Second objective of this article is to introduce a class of connected graphs $\mathfrak{F^{'}}$ such that for each $G$ in this collection, the matrices in $\mathbb{T}_G$ have nonnegative real part up to diagonal unitary similarity. We extend some of the bounds for the spectral radius of the $\mathbb{T}$-gain adjacency matrices of graphs in this class $\mathfrak{F^{'}}$, which are known for Hermitian adjacency matrices.

The spectral radius of the adjacency matrix $ A(G) $ of a graph $ G $ is the largest eigenvalue of $A(G)$. If we consider directed graphs or weighted graphs with complex numbers as edge weights, then the spectral radius need not be an eigenvalue of the associated adjacency matrix. For the Hermitian adjacency matrix of a digraph, surprisingly, in [@Bojan Theorem 5.6] the authors established the following bounds for the spectral radius of the Hermitian adjacency matrix $ H(X) $ in terms of the largest eigenvalue $\lambda_{1}(H(X))$ of $H(X)$.

[@Bojan Theorem 5.6]\[Theo.6\] For every digraph $X$, $ \lambda_{1}(H(X))\leq \rho(H(X))\leq3\lambda_{1}(H(X))$.

Third objective of this article is to extend the above result for $\mathbb{T}$-gain adjacency matrices. For any connected graph, we show that if the associated $\mathbb{T}$-gain graph has nonnegative real part, then the above bounds hold true\[Theorem \[Theo.5.1\]\]. Also, we have for any graph $G \in \mathfrak{F^{'}}$, the bounds in the above theorem hold true irrespective of the gains of the edges\[Theorem \[Theo.5.2\]\].

The spectral radius of the adjacency matrix of a graph is bounded above by the maximum vertex degree of the underlying graph. In [@Bojan], this bound was extended to Hermitian adjacency matrices. Also the sharpness of the bound has been characterized (See Theorem \[Theo.2.6\] of Section 2). Our fourth objective is to introduce the notion of $ k $-generalized Hermitian adjacency matrix, $ H_{k}(X) $ of a digraph $ X $, which extends the notion of Hermitian adjacency matrix of $X$. For this new class of matrices, we characterize the structure of a digraph $ X $ for which the spectral radius $ \rho(H_{k}(X))$ equals to the largest vertex degree of $\Gamma(X)$, which gives an alternative proof of .

Notation, definitions and known results
=======================================

Let $ G=(V(G),E(G)) $ be a simple undirected graph with vertex set $ V(G)=\{v_{1}, v_{2},\dots, \\v_{n} \} $, and edge set $ E(G) $. If the vertex $ v_{i} $ is adjacent to the vertex $ v_{j} $, then we write $ v_{i} \sim v_{j}$. The undirected edge between the vertices $ v_{i} $ and $ v_{j} $, if exists, is denoted by $ e_{ij} $. The *adjacency matrix* of a simple graph $ G $, denoted by $ A(G) $, is the symmetric $ n \times n $ matrix whose $ (i,j)th $ entry is defined by $ a_{ij}=1 $ if $ v_{i} \sim v_{j} $, and $ a_{ij} = 0 $ otherwise .

A digraph is said to be an *oriented graph* if it has no digons. A *mixed graph* is a graph which may contain both directed and undirected edges. When we consider Hermitian adjacency matrix of a mixed graph, the undirected edges are treated as digons. From this point of view, digraphs and mixed graphs are equivalent.

Suppose $ V(T)=\{v_{1}, v_{2}, \dots, v_{n} \} $ is the vertex set of a tree $ T $. Then for $ 0\leq i , j \leq n $, the *undirected path* in $ T $ between the vertices $ v_{i} $ and $ v_{j} $ is denoted by $ v_{i}Tv_{j}:=v_{i}e_{i (i+1)}v_{i+1}e_{(i+1) (i+2)}\dots v_{j-1}e_{(j-1) j}v_{j} $. Similarly, $\overrightarrow{v_{i}Tv_{j}}:=v_{i}\overrightarrow{e_{i(i+1)}}v_{i+1}\overrightarrow{e_{(i+1) (i+2)}}\dots\break v_{j-1}\overrightarrow{e_{(j-1) j}}v_{j}  $ denote the *directed path* in $ T $ from the vertex $ v_{i} $ to the vertex $ v_{j}$. A *rooted tree* is a tree in which one vertex has been fixed as the root. Let $T$ be a rooted tree with the vertex $v_{r}$ as the root. The vertex set of the rooted tree $T$ admits a canonical partial ordering on it as follows: $ v_{x} \leq v_{y}$ if the vertex $ v_{x} $ lies in the path $ v_{r}Tv_{y} $. This partial order is called the *tree-order* on $ V(T) $ associated with the rooted tree $ T $ with root vertex $ v_{r} $ [@Diestel]. A rooted spanning tree $ T $ of a connected graph $ G $ is said to be a *normal spanning tree* if any two adjacent vertices of $G $ are comparable with respect to the tree ordering. Whenever we consider $ T $ as a normal spanning tree of a connected graph $ G $, we assume that $ T $ is a rooted tree with some vertex as its root.

[@Diestel Proposition 1.5.6] \[Theo.1\] Let $ G $ be a connected graph. Then $ G $ has a normal spanning tree with any specified vertex as its root.

Let $ G $ be a connected graph with spanning tree $ T $. Then for each edge $ e \in  E(G)\setminus E(T) $, adding the edge $e$ to $T$ creates a unique cycle in $T \cup \{e\}$. This cycle is called a *fundamental cycle* of $ G $. Let $ \overrightarrow{G} $ be an oriented graph after assigning some direction for each edge of $ G $. Suppose $
    (\overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots, \overrightarrow{e_{m}}) $ is an ordering of the directed edges of $ \overrightarrow{G} $. The *incidence matrix* of $ \overrightarrow{G} $ is an $ m \times n $ matrix whose rows and columns are indexed by vertices and edges of $ \overrightarrow{G} $, respectively, is denoted by $ Q(\overrightarrow{G}) $, and is defined as follows: $$(i,j)th\text{ entry of $ Q(\overrightarrow{G})$}=q_{ij}=\left\{
    \begin{array}{rl}
    0 & \text{if the vertex $ v_{i} $ and the edge $ \overrightarrow{e_{j}}$ are not incident, } \\
    1 & \text{if the edge $ \overrightarrow{e_{j}} $ originates at the vertex $ v_{i},$  }\\
    -1 & \text{if the edge $ \overrightarrow{e_{j}} $ terminates at the vertex $ v_{i}.$  }
    \end{array}\right.$$ The null space of $  Q(\overrightarrow{G}) $ is known as the *cycle subspace* of $ G $. Let $ S $ be a subgraph of $ G $. The *incidence vector* of an oriented subgraph $ \overrightarrow{S} $ with respect to the ordering of the edges $ (\overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots, \overrightarrow{e_{m}}) $ is an $ m \times 1 $ vector defined as follows: If the edge $ e_{i} \in E(G)$ is in $ \overrightarrow{S} $ with its direction is same as that of $ \overrightarrow{e_{i}} $ (opposite to that of $ \overrightarrow{e_{i}} $, respectively) in $ \overrightarrow{G} $, then the $ i$th entry is $ 1$ ($-1$, respectively), and $ 0 $ otherwise. Then the set of incidence vectors of all fundamental cycles associated with $ T $ forms a basis for the cycle subspace of $ G $ [@Bapat Theorem 5.1].

In this article, we call $ \overrightarrow{e_{st}} $ as a directed edge in the context of digraphs, and we call the same as an oriented edge in the context of gain graphs. For any simple graph $ G $, each undirected edge $ e_{st} \in E(G) $ is associated with a pair of oriented edges, namely $ \overrightarrow{e_{st}} $ and $ \overrightarrow{e_{ts}} $. Set of all such oriented edges of a simple graph $ G $ is known as *oriented edge set* of $ G $, and is denoted by $ \overrightarrow{E}(G) $. A *$\mathbb{T}$-gain graph (or complex unit gain graph)* on a simple graph $ G $ is a triple $\Phi=(G,\mathbb{T},\varphi)$ such that the map (*the gain function*) $\varphi:\overrightarrow{E}(G)\rightarrow\mathbb{T}$ satisfies $\varphi(\overrightarrow{e_{st}})=\varphi(\overrightarrow{e_{ts}})^{-1}$. That is, for an oriented edge $\overrightarrow{e_{st}}$, if we assign a value $g$ (the *gain* of the edge $ \overrightarrow{e_{st}}$) from $\mathbb{T}$, then assign $g^{-1 }$ to the oriented edge $\overrightarrow{e_{ts}}$. For simplicity, we use $\Phi=(G,\varphi)$ to denote a $\mathbb{T}$-gain graph instead of $\Phi=(G,\mathbb{T},\varphi)$. We call $ \varphi $ as a $ \mathbb{T}$-gain on $ G $ if $ \Phi=(G, \varphi) $ is a $ \mathbb{T} $-gain graph on $ G $. In [@reff1], the author studied the notion of the adjacency matrix $A(\Phi)=(a_{st})_{n\times n}$ of a $\mathbb{T}$-gain graph $\Phi$. The entries of $A(\Phi)$ are given by $$a_{st}=\begin{cases}
    \varphi(\overrightarrow{e_{st}})&\text{if } \mbox{$v_s\sim v_t$},\\
    0&\text{otherwise.}\end{cases}$$ It is clear that the matrix $A(\Phi)$ is Hermitian, and hence its eigenvalues are real. When $\varphi(\overrightarrow{e_{st}})=1$ for all $\overrightarrow{e_{st}}$, then $A(\Phi)=A(G)$. Thus we can consider $G$ as a $\mathbb{T}$-gain graph and we write this by $(G,1)$. For a square matrix $ B $ with complex entries, $\sigma(B)$ and $\rho(B)$ denote the spectrum and the spectral radius of $ B $, respectively.

A cycle is called [*directed cycle*]{} if its all edges are oriented in same direction. A directed cycle obtained from a cycle $C$ is denoted by $ \overrightarrow{C} $. For any cycle, there are only two directed cycles associated with it. Let $ \overrightarrow{C} \equiv v_{1} \overrightarrow{e_{12}}v_{2}\overrightarrow{e_{23}} \dots v_{k}\overrightarrow{e_{k1}}v_{1} $ be a directed cycle in a $ \mathbb{T}$-gain graph $ \Phi=(G, \varphi) $, then the gain of this cycle, denoted by $\varphi(\overrightarrow{C})$, is defined as $\varphi(\overrightarrow{C}) := \varphi(\overrightarrow{e_{12}})\varphi(\overrightarrow{e_{23}}) \cdots \varphi(\overrightarrow{e_{k1}}) $. If $\varphi(\overrightarrow{C})$ =1, we call the underlying cycle $ C $ as neutral in $ \Phi $. A $\mathbb{T} $-gain graph $ \Phi $ is said to be *balanced* if all the cycles in $ G $ are neutral in $ \Phi $.

Any function from the vertex set of $G$ to the complex unit circle $\mathbb{T}$ is called a *switching function*. Two $\mathbb{T}$-gain graphs $\Phi_1=(G,\varphi_1)$ and $\Phi_2=(G,\varphi_2)$ are said to be *switching equivalent*, denoted by $\Phi_1\sim\Phi_2$, if there is a switching function $\zeta:V(G)\rightarrow\mathbb{T}$ such that $$\varphi_2(\overrightarrow{e_{st}})=\zeta(v_s)^{-1}\varphi_1(\overrightarrow{e_{st}})\zeta(v_t).$$

The switching equivalence of two $\mathbb{T}$-gain graphs can be defined in the following equivalent way: Two $\mathbb{T}$-gain graphs $\Phi_1=(G,\varphi_1)$ and $\Phi_2=(G,\varphi_2)$ are switching equivalent, if there exists a diagonal matrix $D_\zeta$ with diagonal entries from $\mathbb{T}$ such that $$A(\Phi_2)=D_\zeta^{-1}A(\Phi_1)D_\zeta.$$

[@Reff2016 Lemma 2.2]\[Theo.3\] Let $\varphi_{1}$ and $ \varphi_{2}$ be two $ \mathbb{T}$-gains on a connected graph $ G $. If for every cycle $ C $ in $ G $ there is a directed cycle $ \overrightarrow{C} $ such that $ \varphi_{1}(\overrightarrow{C})=\varphi_{2}(\overrightarrow{C}) $, then $ \Phi_{1} \sim \Phi_{2} $.

The following result gives an upper bound for the spectral radius of $A(\Phi)$ in terms of the maximum vertex degree $\Delta$ of $G$.

\[Theo.12\] Let $ \varphi $ be a $ \mathbb{T} $-gain on a graph $ G $. Then $ \rho(A(\Phi)) \leq \Delta. $

Next, we collect a couple of results related to the spectrum and the spectral radius of adjacency matrices of $\mathbb{T}$-gain graphs.

[@Our-gain1 Theorem 4.6] \[Theo.7\] Let $ \varphi $ be a $ \mathbb{T} $-gain on a connected graph $ G $. Then, $ \sigma(A(\Phi)) =\sigma(A(G))$ if and only if $ \Phi $ is balanced.

[@Our-gain1 Lemma 4.1] \[Theo.8\] Let $ \varphi $ be a $ \mathbb{T} $-gain on a connected graph $ G $, then $\rho(A(\Phi)) \leq \rho(A(G)) $.

[@Our-gain1 Theorem 4.4]\[Theo.9\] Let $ \varphi $ be a $ \mathbb{T} $-gain on a connected graph $ G $, then $\rho(A(\Phi))=\rho(A(G))$ if and only if either $\Phi$ or $ -\Phi $ is balanced.

The *characteristic polynomial* of an $n \times n$ matrix $ M $ is defined as $ det(M-xI) $, where $ I $ is the $n \times n$ identity matrix. A graph $G$ is called an *elementary graph*, if each of its component is either an edge or a cycle. Let $\mathcal{H}(G)$ denote the collection of all spanning elementary subgraphs of a graph $G$. For any $H \in\mathcal{H}(G)$, let $\mathcal{C}(H)$ denote the collection of all cycles in $H$.

\[Theo4\] Let $\Phi$ be any $\mathbb{T}$-gain graph with the underlying graph $G$. Let $P_\Phi(x)=x^n+a_1x^{n-1}+\cdots+a_n$ be the characteristic polynomial of $A(\Phi)$. Then $$a_i=\sum_{H\in\mathcal{H}_i(G)}(-1)^{p(H)}2^{c(H)}\prod_{C\in \mathcal{C}(H)}\Re(\varphi(C)),$$ where $\mathcal{H}_i(G)$ is the set of all elementary subgraphs of $G$ with $i$ vertices. $ p(H) $ and $ c(H) $ are the number of components and the number of cycles in $ H $, respectively. $\Re(\varphi(C)) $ is the real part of the gain of a directed cycle $ \overrightarrow{C}$.

Next result gives a bound for the spectral radius of the adjacency matrix $A(G)$ of a simple graph $G$ in terms of the largest vertex degree of $G$.

\[Theo.11\] Let $ G $ be a connected graph with $ n $ vertices and $ m $ edges. Then $ \rho(G) \leq \Delta$. Furthermore, equality holds if and only if $ G $ is regular.

A digraph is said to be *weakly connected* if $ \Gamma(X) $ is connected. The following extension of the above result for the Hermitian adjacency matrices is known.

\[Theo.2.6\] Let $ X $ be a digraph and $ \Delta $ be the largest vertex degree of $ \Gamma(X) $. Then $ \rho(H(X))\leq \Delta $. If $ X $ is weakly connected, then equality holds if and only if $ \Gamma(X) $ is $ \Delta $-regular and there is a partition of vertex set of $ X $ into four parts( possibly empty) $ V_{1}, V_{-1}, V_{i} $, and $ V_{-i} $ such that one of the following holds:

1.  For each $ j \in \{ \pm 1, \pm i\} $, the digraph induced by $ V_{j} $ in $ X $ contains only digons. Every other directed edge $ \overrightarrow{e_{st}} $ of $ X $ such that $ v_{s} \in V_{j} $ and $ v_{t} \in V_{(-i)j} $ for some $ j \in \{ \pm 1, \pm i\} $. See Figure \[fig1.1\].

2.  For each $ j \in \{ \pm 1, \pm i\} $, the digraph induced by $ V_{j} $ in $ X $ is an independent set. For each $ j \in \{ \pm 1, \pm i\} $, every directed edge with one end in $ V_{j} $ and one end in $ V_{-j} $ is contained in a digon. Every other directed edge $ \overrightarrow{e_{st}} $ of $ X $ is such that $ v_{s}\in V_{j} $ and $ v_{t} \in V_{ij} $ for some $ j \in \{ \pm 1, \pm i\} $. See Figure \[fig1.1\].

\

Let $ A$ be an $n \times n$ Hermitian matrix and let the eigenvalues of $ A $ be ordered as $\lambda_{min}=\lambda_n\leq\lambda_{n-1}\leq \ldots \leq\lambda_1=\lambda_{max}$. Then $ \lambda_{min} \leq x^{*}Ax \leq \lambda_{max} $ for any vector $ x \in \mathbb{C} ^{n}$ with $x^{*}x=1$, with equality in the right-hand(respectively, left-hand) inequality if and only if $ Ax=\lambda_{max} x$ (respectively , $ Ax=\lambda_{min}x $). Moreover,

$ \lambda_{max}= \max_{x \neq 0}\frac{x^{*}Ax}{x^{*}x} $ and $ \lambda_{min}= \min_{x \neq 0}\frac{x^{*}Ax}{x^{*}x} $.

The *numerical range* of an $n \times n$ complex matrix $A$ is a subset of the complex numbers $ \mathbb{C} $, defined as follows:

$ W(A):=\{\langle Ax, x \rangle: x \in \mathbb{C}^n, \langle x, x \rangle=1  \} $.

For any $n \times n $ complex matrix $A$, the numerical range $W(A)$ is a convex set.

[@nrange] \[Theo.5\] If $A$ is an $n \times n$ Hermitian matrix, then $ W(A)$ is an interval $ [m,M] $, where $ m=\lambda_{min}(A), M=\lambda_{max}(A) $. Moreover, $ \rho(A) =   \max \{|m|,|M|\} $.

This article is organized as follows: In section 3, we study the cospectrality of matrices in $ \mathbb{T}_{G} $. First, we introduce an orientation scheme for the edges of the graph $G$, using a normal spanning tree of $G$, and we call the resultant oriented graph as suitably oriented graph. Suitably oriented graph plays an important role in this article. Using this notion, we construct uncountable many matrices in $ \mathbb{T}_{G} $ having the same spectrum. Then we give an equivalent condition for a graph $ G $ to be a tree in terms of the spectrum and the spectral radius of matrices in $ \mathbb{T}_{G} $. Next, we give a class of connected graphs such that for each graph $ G $ in this class, the cospectrality of matrices in $ \mathbb{T}_{G} $ is characterized in terms of the real part of gains of their cycles.

In section 4, we introduce a new class of connected graphs, $ \mathfrak{F^{'}} $ in terms of $ \mathbb{T}$-gains. More precisely, if $ G \in \mathfrak{F^{'}}$, then the matrices in $ \mathbb{T}_{G} $ have nonnegative real part up to diagonal unitary similarity. In section 5, we discuss bounds for the spectral radius of matrices in $ \mathbb{T}_{G} $. First, we obtain bounds of the spectral radius in terms of the largest eigenvalue of matrices in $ \mathbb{T}_{G} $, where $ G \in \mathfrak{F^{'}} $. Next, we establish a characterization for the gain graph $ \Phi = (G, \varphi) $ to have $ \rho(A(\Phi)) =\Delta $. Then, for any digraph $ X $, we define the notion of $k$-generalized Hermitian adjacency matrix, $ H_{k}(X) $, which is an extension of the notion of Hermitian adjacency matrix. Finally, we characterize digraph $ X $ such that $ \rho(H_{k}(X)) =\Delta $.

Cospectral $ \mathbb{T} $-gain graph {#co-spec}
====================================

For any simple graph $ G $, the collection of all $ \mathbb{T} $-gain graphs associated with $ G $ is denoted by $\mathcal{T}_{G}$. Let us consider $ \mathbb{T}_{G}:=\{A(\Phi): \Phi \in \mathcal{T}_{G} \} $. The spectrum of $ A(\Phi) $ is called the *$\mathbb{T}$-spectrum* of $ \Phi $, and the spectral radius of $A(\Phi)$ is called $\mathbb{T}$-spectral radius of $\Phi$. Let $ \Phi_{1}=(G, \varphi_{1})$ and $ \Phi_{2}=(G, \varphi_{2}) $ be two $ \mathbb{T}$-gain graphs having the same $ \mathbb{T}$-spectrum. Then $ \Phi_{1} $ and $ \Phi_{2} $ are called *$ \mathbb{T}$-cospectral*.

A considerable amount of literature is available on the construction of cospectral simple graphs. From the $ \mathbb{T}$-gain point of view, we may ask the same question about the class of $ \mathbb{T} $-cospectral gain graphs on $ G $.

\

Let us first consider the example given in Figure \[fig1\]. From the figure, we have $ \sigma (A(\Phi_{1}))=\{-1.9318, -0.5176, 0.5176, 1.9318 \} $ and $ \sigma (A(\Phi_{2}))   =\{ 1.4811, -1, 0.3111, 2.1701\} $. Thus, in spite of having the same underlying graph $ G $, $ A(\Phi_{1}) $ and $A(\Phi_{2}) $ have different spectrum.

In this section, we define the notion of suitably oriented graph, and using this notion we construct uncountable many matrices in $ \mathbb{T}_{G} $ having the same spectrum.

\[def.1\] Let $ G $ be a connected graph with vertex set $ V(G)=\{v_{1}, v_{2}, \dots, v_{n} \} $. Let $ T $ be a normal spanning tree of $ G $. A *suitably oriented graph* of a connected graph $ G $ with respect to $ T $ is an oriented graph, denoted by $ \overrightarrow{G_{T}} $ and is defined by assigning orientation on each edge of $ G $ as follows:\
$ \overrightarrow{e_{st}} \in E(\overrightarrow{G_{T}})$ $ v_{s} \leq v_{t} $ and $ e_{st} \in E(T) $, and\
$ \overrightarrow{e_{ts}} \in E(\overrightarrow{G_{T}})$ $ v_{s} \leq v_{t} $ and $ e_{st} \in E(G)\setminus E(T)$, where $ \leq  $ denote the tree order.

Let $ T $ be a spanning tree of a connected graph $ G $ with $ n $ vertices and $ m $ edges. Let $ C_{1}(T), C_{2}(T), \dots , C_{m-n+1}(T) $ be the fundamental cycles of $ G $ with respect to $ T $. If $ T $ is a normal spanning tree, then by definition \[def.1\], it is clear that all the cycles $ C_{1}(T), C_{2}(T), \dots , C_{m-n+1}(T) $ in $ G $ are the directed cycles in $ \overrightarrow{G_{T}} $. We denote them by $ \overrightarrow{C_{1}(T)}, \overrightarrow{C_{2}(T)},\dots, \overrightarrow{C_{m-n+1}(T)} $, and call them as *directed fundamental cycles* in $ \overrightarrow{G_{T}} $.

For any $ \mathbb{T} $-gain graph $ \Phi=(G, \varphi) $, if $ \varphi(\overrightarrow{e_{k}}) =e^{i\theta_{k}}$ for all $ 1 \leq k \leq m$ and $( \overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots, \overrightarrow{e_{m}})$ is an ordering of the directed edges of $ \overrightarrow{G_{T}} $, then we can associate a unique $m$-vector $ \theta(\Phi):= (\theta_{1}, \theta_{2}, \dots, \theta_{m})$ with $ \Phi $.

            

\[Theo3.1\] Let $ \varphi_{1} $ and $  \varphi_{2} $ be two $ \mathbb{T} $-gains on a connected graph $ G $ with $ n $ vertices and $ m $ edges. Let $ T $ be a normal spanning tree of $ G $ and $ \overrightarrow{C_{j}(T)} $, $ j=1,2, \dots, (m-n+1) $ be the directed fundamental cycles in $ \overrightarrow{G_{T}} $. Then $ \varphi_{1}(\overrightarrow{C_{j}(T)} )= \varphi_{2}(\overrightarrow{C_{j}(T)} ) $, for all $ j=1,2, \dots (m-n+1) $ if and only if $ \varphi_{1}(\overrightarrow{C})=\varphi_{2}(\overrightarrow{C}) $, for any cycle $ C $ in $ G $.

Let $G$ be a connected graph with $n$ vertices and $m$ edges. By Theorem \[Theo.1\], $G$ contains a normal spanning tree $ T $ with specified vertex $ r $ as its root. Let $ \overrightarrow{C_{1}(T)}, \overrightarrow{C_{2}(T)}, \dots,$ $ \overrightarrow{C_{m-n+1}(T)} $ be the directed fundamental cycles in $ \overrightarrow{G_{T}} $. For $1 \leq k \leq m$, let $ \overrightarrow{e_{k}} $ denote the directed edges of $\overrightarrow{G_{T}} $, and let $( \overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots, \overrightarrow{e_{m}})$ be an ordering of the directed edges of $ \overrightarrow{G_{T}} $. Let $Q$ be the incidence matrix of $ \overrightarrow{G_{T}} $ associated with $( \overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots, \overrightarrow{e_{m}})$. For any cycle $ C $ in $ G $, let $ I(\overrightarrow{C}) $ denotes the incidence vector of a directed cycle $ \overrightarrow{C}$ with respect to $Q$. Then $ I(\overrightarrow{C}) $ is in the null space of $ Q $. Since the set of all incidence vectors of the directed fundamental cycles in $ \overrightarrow{G_{T}} $ forms a basis for the cycle subspace of $ G $, $ I(\overrightarrow{C}) $ can be expressed as $I(\overrightarrow{C})=c_{1}I(\overrightarrow{C_{1}(T)})+c_{2}I(\overrightarrow{C_{2}(T)})+ \dots+ c_{m-n+1}I(\overrightarrow{C_{m-n+1}(T)}) $, for some real numbers $c_1, c_2, \dots, c_{m-n+1}$.

The gain of any directed cycle $ \overrightarrow{C} $ in $ \Phi $ is $ \varphi({\overrightarrow{C}})= e^{i\langle  I(\overrightarrow{C}),  \theta(\Phi)\rangle}$, where $ \langle,\rangle $ denotes the usual inner product. Let $ \theta(\Phi_{1}) =(\alpha_{1}, \alpha_{2}, \dots, \alpha_{m})$ and $ \theta(\Phi_{2}) =(\beta_{1}, \beta_{2}, \dots, \beta_{m})$. If $ \overrightarrow{C}$ is any directed cycle, then, $$\begin{aligned}
    \varphi_{1}(\overrightarrow{C}) &=& e^{i\langle  I(\overrightarrow{C)}, \theta(\Phi_{1})\rangle} \\
        &=& e^{i \langle c_{1}I(\overrightarrow{C_{1}(T)})+c_{2}I(\overrightarrow{C_{2}(T)})+ \dots+ c_{m-n+1}I(\overrightarrow{C_{m-n+1}(T)}), \theta(\Phi_{1}) \rangle }\\        
    &=& e^{{ c_{1} i \langle I(\overrightarrow{C_{1}(T)}) , \theta(\Phi_{1}) \rangle }+{c_{2} i \langle I(\overrightarrow{C_{2}(T)}) , \theta(\Phi_{1}) \rangle }+ \dots +{c_{m-n+1} i\langle  I(\overrightarrow{C_{m-n+1}(T)}) , \theta(\Phi_{1})\rangle }}\\
        &=&  \left({e^{i \langle  I(\overrightarrow{C_{1}(T)}) , \theta(\Phi_{1}) \rangle  }} \right) ^{c_{1}} \left( {e^{i  \langle I(\overrightarrow{C_{2}(T)}) , \theta(\Phi_{1}) \rangle }}\right)^{c_{2}}\dots \left( {e^{i \langle  I(\overrightarrow{C_{m-n+1}(T)}) , \theta(\Phi_{1}) \rangle  }}\right) ^{c_{m-n+1}}  \\
        &=&  \left({e^{i \langle  I(\overrightarrow{C_{1}(T)}) , \theta(\Phi_{2}) \rangle  }} \right) ^{c_{1}} \left( {e^{i  \langle I(\overrightarrow{C_{2}(T)}) , \theta(\Phi_{2}) \rangle }}\right)^{c_{2}}\dots \left( {e^{i \langle  I(\overrightarrow{C_{m-n+1}(T)}) , \theta(\Phi_{2}) \rangle  }}\right) ^{c_{m-n+1}}  \\
        &=& \varphi_{2}(\overrightarrow{C}).
    \end{aligned}$$

Let $ \varphi_{1} $ and $ \varphi_{2} $ be two $ \mathbb{T}$-gains on a connected graph $ G $ with $ n $ vertices and $ m $ edges. If $ \varphi_{1}( \overrightarrow{C_{k}(T)})=\varphi_{2}( \overrightarrow{C_{k}(T)}) $ for $ k=1, 2, \dots, (m-n+1) $ holds for a normal spanning tree $ T $, then this holds for any normal spanning tree of $ G $.

Therefore, in corollary \[Theo3.1\], the choice of normal spanning tree is immaterial. The next corollary proves that the gains of the directed fundamental cycles determine whether two $ \mathbb{T} $-gain graphs on a same underlying graph are switching equivalent or not.

\[cor.3.2\] Let $ \varphi_{1} $ and $ \varphi_{2} $ be two $ \mathbb{T} $-gains on a connected graph $ G $ with $ n $ vertices and $ m $ edges, and $T$ be any normal spanning tree of $ G $. Then $ \Phi_{1} \sim \Phi_{2} $ if and only if $ \varphi_{1}( \overrightarrow{C_{k}(T)})=\varphi_{2}( \overrightarrow{C_{k}(T)}) $, $k=1,2,\dots, (m-n+1) $.

If $ \Phi_{1} \sim \Phi_{2} $, then it is clear that $ \varphi_{1}( \overrightarrow{C_{k}(T)})=\varphi_{2}( \overrightarrow{C_{k}(T)}) $, $ k=1,2,\dots, (m-n+1) $. Proof of the converse follows from Theorem \[Theo3.1\] and Theorem \[Theo.3\].

Here we are giving an algorithm to check whether two $ \mathbb{T} $-gains are switching equivalent or not.

  ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  [**Algorithm 1.**]{} Identify whether $ \Phi_{1} \sim \Phi_{2} $ or not.
  ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
  1: Compute a normal spanning $T $.

  2: Compute $( \overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots,
       \overrightarrow{e_{m}})$, the edge ordering with respect to $ T $.

  3: Two gains $ \theta(\Phi_{1}) =(\theta_{1}, \theta_{2}, \dots, \theta_{m})$ and $ \theta(\Phi_{2}) =(\psi_{1}, \psi_{2}, \dots, \psi_{m})$.

  4: Compute the incidence vectors $ I(\overrightarrow{C_{j}(T)}), j=1,2, \dots, (m-n+1) $, where $ \overrightarrow{C_{j}(T)} $ are

      the directed fundamental cycles with respect to $T$.

  5: Compute $ \varphi_{1}(\overrightarrow{C_{j}(T)})= e^{i \langle I(\overrightarrow{C_{j}(T)}),  \theta(\Phi_{1}) \rangle }$ and $ \varphi_{2}(\overrightarrow{C_{j}(T)})= e^{i \langle I(\overrightarrow{C_{j}(T)}),  \theta(\Phi_{2}) \rangle }$

      for all $ j=1,2, \dots, (m-n+1)$.

  6: Check $ \varphi_{1}(\overrightarrow{C_{j}(T)}) =\varphi_{2}(\overrightarrow{C_{j}(T)})$ for $ j=1,2, \dots (m-n+1) $.

  7: [**Return:** ]{} If step $(6)$ true, then $ \Phi_{1} \sim \Phi_{2} $; otherwise $ \Phi_{1} \nsim \Phi_{2} $.
  ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------

Proof of the following theorem is immediate from corollary \[cor.3.2\]. We give a proof of the same with out using corollary \[cor.3.2\].

\[Theo3.2\] Let $ \varphi_{1} $ and $ \varphi_{2} $ be two $ \mathbb{T}$-gains on a connected graph $ G $ with $ n $ vertices and $ m $ edges. Let $T$ be a normal spanning tree of $ G $ such that $\varphi_{1}(\overrightarrow{C_{j}(T)})=\varphi_{2}(\overrightarrow{C_{j}(T)}) $, for all $ j=1,2,\dots, (m-n+1) $. Then, $A(\Phi_1)$ and $A(\Phi_2)$ have the same spectrum. i.e., $\Phi_1$ and $\Phi_2$ are $\mathbb{T}$-cospectral.

By Theorem \[Theo3.1\], $ \varphi_{1}(\overrightarrow{C}) = \varphi_{2}(\overrightarrow{C}) $, for any cycle $ C $ in $ G $. There are exactly two directed cycles can be defined on an undirected cycle $ C $. Gains of these two cycles are conjugate to each other. Since the real part of their gains is independent of choice of directed cycles, so instead of writing $ \Re( \varphi_{1}(\overrightarrow{C})) $, we write $ \Re( \varphi_{1}(C)) $. Therefore, for any cycle $ C $ in $ G $, we have $ \Re(\varphi_{1}(C))=\Re(\varphi_{2}(C)) $.

By Theorem \[Theo4\], for any $ \mathbb{T} $-gain graph $ \Phi=(G, \varphi) $, the coefficients of the characteristic polynomial $P_\Phi(x)=x^n+a_1x^{n-1}+\cdots+a_n$ of $ A(\Phi) $ have the following expression: $$a_i=\sum_{H\in\mathcal{H}_i(G)}(-1)^{p(H)}2^{c(H)}\prod_{C\in \mathcal{C}(H)}\Re(\varphi(C)).$$ Thus, $P_{\Phi_{1}}(x)=P_{\Phi_{2}}(x)$, and hence $A(\Phi_1)$ and $A(\Phi_2)$ have the same spectrum.

Let $ \mathbb{I}:=[0, 2\pi)  \subseteq \mathbb{R}$. Let $ \varphi $ be a $ \mathbb{T} $-gain on a connected graph $ G $. For any fixed normal spanning tree $ T $ of $ G $, let us consider a collection of $ \mathbb{T} $-gain graphs as below.\
For any $ r = (c_{1}, c_{2}, \dots, c_{m-n+1}) \in \mathbb{I}^{m-n+1} $, $$\mathcal{A}_{T}(r):=\{\Phi=(G, \varphi): \varphi(\overrightarrow{C_{j}(T)})= e^{ic_{j}},  j=1,2, \dots , (m-n+1)\}.$$

\[lem.3.1\] Let $ G $ be a connected graph with $ n $ vertices and $ m $ edges. Then for each $ r = (c_{1}, c_{2}, \dots , c_{m-n+1}) \in \mathbb{I}^{m-n+1} $, $ \mathcal{A}_{T}(r) \subseteq \mathcal{T}_{G}$ contains uncountable collection of switching equivalent $ \mathbb{T} $-gain graphs. That is, all the $ \mathbb{T} $-gain graphs in $ \mathcal{A}_{T}(r)  $ are $ \mathbb{T}$-cospectral.

Using corollary \[cor.3.2\], it is clear that $ \mathcal{A}_{T}(r)  $ is a collection of switching equivalent $ \mathbb{T} $-gain graphs. Let $ T $ be a normal spanning tree of a connected graph $ G $. Let $ \overrightarrow{T}$ be the oriented tree in $ \overrightarrow{G}_{T} $, where the orientation is obtained from $\overrightarrow{G}_{T}$. Suppose $ E(\overrightarrow{G}_{T}\setminus\overrightarrow{T}) =\{ \overrightarrow{e_{1}}, \overrightarrow{e_{2}}, \dots, \overrightarrow{e_{m-n+1}}\} $. Then, each edge $ \overrightarrow{e_{j}} $ produces a unique directed fundamental cycle, say $ \overrightarrow{C_{j}(T)} $. Let $ \varphi $ be an arbitrary $ \mathbb{T} $-gain on $ T $. Define $ \tilde{\varphi} $ on $ G $ as follows:

$$\tilde{\varphi}(\overrightarrow{e})= \left\{ \begin{array}{ll}
    \varphi(\overrightarrow{e}) & \mbox{if $ \overrightarrow{e} \in E(\overrightarrow{T}) $},\\
     \frac{e^{ic_{j}}}{\varphi( \overrightarrow{C_{j}(T)}\setminus \overrightarrow{e_{j}})} & \mbox{if $  \overrightarrow{e}=\overrightarrow{e_{j}} \in E( \overrightarrow{G}_{T}\setminus\overrightarrow{T})$ for some $ j. $}
     \end{array} \right.$$

It is easy to see that $ \tilde{\Phi}=(G, \tilde{\varphi}) \in  \mathcal{A}_{T}(r) $. Since $ \varphi $ is arbitrary, so $\mathcal{A}_{T}(r) $ is uncountable.

Let us consider $ G=K_{4} $ with vertex set $ \{v_{1}, v_{2}, v_{3}, v_{4} \} $. Consider a normal spanning tree $ T $ of $ K_{4} $ as a path on four vertices, $ T \equiv v_{1}e_{12}v_{2}e_{23}v_{3}e_{34}v_{4} $. The directed fundamental cycles with respect to $ T $ are $ \overrightarrow{C_{1}(T)} \equiv v_{1}\overrightarrow{e_{12}}v_{2}\overrightarrow{e_{23}}v_{3}\overrightarrow{e_{31}}v_{1}, \overrightarrow{C_{2}(T)} \equiv v_{1}\overrightarrow{e_{12}}v_{2}\overrightarrow{e_{23}}v_{3}\overrightarrow{e_{34}}v_{4}\overrightarrow{e_{41}}v_{1} $ and $\overrightarrow{C_{3}(T)} \equiv v_{2}\overrightarrow{e_{23}}v_{3}\overrightarrow{e_{34}}v_{4}\overrightarrow{e_{42}}v_{2}$ in $ \overrightarrow{G}_{T}$. Let $ r=(\pi, \frac{\pi}{2}, \frac{3\pi}{2}) \in  \mathbb{I}^{3}$. Then, by Lemma \[lem.3.1\], $ \mathcal{A}_{T}(r) $ contains uncountably many $ \mathbb{T} $-cospectral gain graphs. Some of them are given in Figure \[fig17\] .

\

In the next result, we show that the collection $ \mathcal{A}_{T}(r) $ partition $ \mathcal{T}_{G} $.

Let $ T $ be a normal spanning tree of a connected graph $ G $ with $ n $ vertices and $ m $ edges. Then $ \mathcal{T}_{G}= \bigsqcup_{r\in \mathbb{I}^{m-n+1}}\mathcal{A}_{T}(r)$, where $\bigsqcup$ denote the disjoint union.

Using lemma \[lem.3.1\], it is clear that, for any $ r \in \mathbb{I}^{m-n+1} $, $ \mathcal{A}_{T}(r) \neq \emptyset $. For any $ \Phi \in \mathcal{T}_{G}$, it is easy to see that $\Phi \in \mathcal{A}_{T}(r) $, for some $r \in \mathbb{I}^{m-n+1}$. Let $ r, s \in \mathbb{I}^{m-n+1} $ with $ r \neq s $. Suppose that $ \mathcal{A}_{T}(r) \cap \mathcal{A}_{T}(s) $ is nonempty. Let $ \Phi \in \mathcal{A}_{T}(r) \cap \mathcal{A}_{T}(s)$. Let $ r=(r_{1}, r_{2}, \dots, r_{(m-n+1)}) $ and $ s=(s_{1}, s_{2}, \dots, s_{(m-n+1)}) $. Since $ r \neq s $, so $ r_{k} \neq s_{k} $ for some $k$. Then $ \varphi(\overrightarrow{C_{k}(T)} )=e^{i s_{k}} \neq e^{i r_{k}}= \varphi(\overrightarrow{C_{k}(T)} )$, which is absurd. Thus the collection $ \mathcal{A}_{T}(r) $ partition $ \mathcal{T}_{G} $.

The partition $ \bigsqcup_{r\in \mathbb{I}^{m-n+1}}\mathcal{A}_{T}(r) $ is induced by the switching equivalence relation. That is, the relation is “switching equivalent to”.

We have already seen that the elements of each partition $\mathcal{A}_{T}(r) $ are $\mathbb{T} $-cospectral. Define ${\bf 0} = (0, \dots, 0) \in \mathbb{I}^{m-n+1}.$ Then $ \mathcal{A}_{T}({\bf 0}) $ is the collection of balanced $ \mathbb{T} $-gain graphs. By Theorem \[Theo.7\], it can be seen that $ \mathcal{A}_{T}(\bf 0) $ is not $ \mathbb{T} $-cospectral with any other $ \mathcal{A}_{T}(s) $. Now, let us consider the following question: For any connected graph $ G $ and $ {\bf 0} \neq r  \in \mathbb{I}^{m-n+1}$, does there exist $ s \in \mathbb{I}^{m-n+1}$ such that $r \neq s $, and $\mathcal{A}_{T}(r) $ and $ \mathcal{A}_{T}(s)$ are $ \mathbb{T}$-cospectral ? To answer this, first let us estabilish the following lemma.

\[lemma3.2\] Let $ T $ be a normal spanning tree of a connected graph $ G $ with $ n $ vertices and $ m $ edges. If for any $ r, s \in \mathbb{I}^{m-n+1}$ with $r \neq s $ such that $ \Re(\varphi(C))=\Re(\psi(C)) $ for all cycle $ C $ in $ G $ and for some $ \Phi \in\mathcal{A}_{T}(r)  $ and $ \Psi \in \mathcal{A}_{T}(s) $. Then $\mathcal{A}_{T}(r) $ and $ \mathcal{A}_{T}(s)$ are $ \mathbb{T}$-cospectral.

Proof is a consequence of Theorem \[Theo3.2\].

Using the above lemma, for any $ r \in \mathbb{I}^{m-n+1}$, let us construct $ s\in \mathbb{I}^{m-n+1} $ so that $\mathcal{A}_{T}(r) $ and $ \mathcal{A}_{T}(s)$ are $ \mathbb{T}$-cospectral.

\[lamma3.3\] Let $ T $ be a normal spanning tree of a connected graph $ G $ with $ n $ vertices and $ m $ edges. For any $ {\bf 0} \neq  r=( r_{1}, r_{2}, \dots, r_{m-n+1}) \in \mathbb{I}^{m-n+1} $, consider $ s=(2\pi-r_{1}, 2\pi-r_{2}, \dots, 2\pi-r_{m-n+1}) \in  \mathbb{I}^{m-n+1}$ ($\mod 2 \pi$). Then $\mathcal{A}_{T}(r) $ and $ \mathcal{A}_{T}(s)$ are $ \mathbb{T}$-cospectral.

Let $ \Psi \in \mathcal{A}_{T}(r) $ and $ \Phi \in \mathcal{A}_{T}(s) $. Suppose $ C $ is a cycle in $ G $. Let $I(\overrightarrow{C})=c_{1}I(\overrightarrow{C_{1}(T)})+c_{2}I(\overrightarrow{C_{2}(T)})+ \dots+ c_{m-n+1}I(\overrightarrow{C_{m-n+1}(T)}) $, for some real $c_1, c_2, \dots, c_{m-n+1}$. Then, using Theorem \[Theo3.1\], we have $$\begin{aligned}
    \varphi ( \overrightarrow{C}) &=& \left({e^{i \langle  I(\overrightarrow{C_{1}(T)}) , \theta(\Phi) \rangle  }} \right) ^{c_{1}} \left( {e^{i  \langle I(\overrightarrow{C_{2}(T)}) , \theta(\Phi) \rangle }}\right)^{c_{2}}\dots \left( {e^{i \langle  I(\overrightarrow{C_{m-n+1}(T)}) , \theta(\Phi) \rangle  }}\right) ^{c_{m-n+1}} \\
    &=& \left(e^{i(2\pi-r_{1})} \right) ^{c_{1}}  \left(e^{i(2\pi-r_{2})} \right) ^{c_{2}} \dots  \left(e^{i(2\pi-r_{m-n+1})} \right) ^{c_{m-n+1}} \\
    &=&e^{-i(r_{1}c_{1}+r_{2}c_{2}+\dots+r_{m-n+1}c_{m-n+1})}. \\
    \end{aligned}$$ Similarly, $\psi ( \overrightarrow{C})= e^{i(r_{1}c_{1}+r_{2}c_{2}+\dots+r_{m-n+1}c_{m-n+1})} $. So $ \Re( \varphi (C))=\Re (\psi (C)) $. Hence, by Lemma \[lemma3.2\], $ \mathcal{A}_{T}(r)$ and $ \mathcal{A}_{T}(s)$ are $ \mathbb{T}$-cospectral.

Let $ \{0,\pi\}^{m-n+1}:=\{ (r_{1},r_{2}, \dots, r_{m-n+1}): \text{ $ r_{j}=0 $ or $  \pi $ }\} $. Now, we observe that if $ r \in \mathbb{I}^{m-n+1}\setminus \{0, \pi \}^{m-n+1} $, then for any $ \Phi \in \mathcal{A}_{T}(r) \subseteq \mathcal{T}_{G} $, there exist a $ \Psi \in \mathcal{T}_{G} $ with $ \Phi \nsim \Psi $ have the same $ \mathbb{T} $-spectrum. Next theorem shows that all the classes $\mathcal{A}_{T}(r) $ are $ \mathbb{T} $-cospectral if and only if $ G $ is a tree.

\[spec-equi-tree\] Let $ \Phi=(G,\varphi)$ be a $ \mathbb{T}$-gain graph. Then the following are equivalent:

1.  $G$ is tree.

2.  Any two gain graphs in $ \mathcal{T}_{G} $ are $ \mathbb{T} $-cospectral.

3.  Adjacency matrices of any two gain graphs in $ \mathcal{T}_{G} $ have same spectral radius.

\(1) $\Rightarrow$ (2): Since $ G $ is a tree, for any choice of the gain function $ \varphi $, $ G $ is balanced. Therefore, by Theorem \[Theo.7\], $ \sigma(A(G)) = \sigma ( A(\Phi))$ for all $ A(\Phi) \in \mathbb{T}_{G} $. Hence the $ \mathbb{T} $-spectrum is invariant in $\mathcal{T}_{G} $.\
(2) $\Rightarrow$ (3): Easy to verify.\
(3) $\Rightarrow$ (1): By Theorem \[Theo.9\], we know that, $ \rho(A(G))=\rho(A(\Phi)) $ if and only if either $ \Phi $ or $ -\Phi $ is balanced. Therefore, $ \rho(A(\Phi)) $ is invariant implies that either $ \Phi $ or $ -\Phi $ is balanced for any $ \varphi $. This is possible only when $ G $ is a tree. For, if there is a cycle in $G$, then we construct a $ \mathbb{T} $-gain $\varphi$ on $ G $ so that, neither $ \Phi $ nor $ -\Phi $ is balanced.

We have seen that, for any two $ \mathbb{T} $-gain graphs $\Phi_{1}, \Phi_{2} \in \mathcal{T}_{G}$, if $ \Re(\varphi_{1}(C))= \Re(\varphi_{2}(C)) $ for all cycles $ C $ in $ G $, then they are $ \mathbb{T} $-cospectral. However the converse of this statement need not be true.

\

Consider the graph given in Figure \[fig2\]. Here $ \sigma(A(\Phi_{1}))=\sigma(A(\Phi_{2}))=\{ -2.37, -1.41,\break -0.59, 0, 0.59, 1.41, 2.37 \} $ but $ \Re(\varphi_{1}(C_{i})) \ne \Re(\varphi_{2}(C_{i})) $, $ i=1,2$. Next, we are interested in identifying classes of graphs for which $ \Re(\varphi_{1}(C))= \Re(\varphi_{2}(C)) $ holds for all cycles $ C $ in $ G $ if and only if $\Phi_{1}$ and $\Phi_{2}$ are $ \mathbb{T} $-cospectral.

A connected graph $ G $ is said to have the property *spectrum determined by cycle gains*(SDC), if for any two $ \mathbb{T} $-gains $ \varphi_{1} $ and $ \varphi_{2} $ on $ G $, $ \Re(\varphi_{1}(C))=\Re(\varphi_{2}(C))$ holds for every cycle $ C $ in $ G $ if and only if $ \Phi_{1} $ and $ \Phi_{2} $ are $ \mathbb{T} $-cospectral.

Next, we construct a class of connected graphs which has SDC property.

Let $ \mathcal{D}_{n} $ denote the collection of all connected graphs $ G $ on $ n $ vertices such that for each $ k $, $ 3 \leq k \leq n $, $ G $ has at most one cycle of length $ k $.

In Figure 4, we give two graphs from the class $ \mathcal{D}_{10} $.

\

In the next theorem, we show that elements of $ \mathcal{D}_{n} $ have the SDC property.

Let $ \varphi_{1} $ and $ \varphi_{2} $ be two $ \mathbb{T} $-gains on $ G \in \mathcal{D}_{n} $. Then $\sigma(A(\Phi_{1}))=\sigma(A(\Phi_{2})) $ if and only if $ \Re(\varphi_{1}(C))=\Re(\varphi_{2}(C)) $ for all cycles $ C$ in G.

Let us assume that $ \sigma(A(\Phi_{1}))=\sigma(A(\Phi_{2})) $. Let $ P_{\Phi_{1}}(x)=x^{n}+a_{1}x^{n-1}+ \dots + a_{n} $ and $  P_{\Phi_{2}}(x) =x^{n}+b_{1}x^{n-1}+ \dots + b_{n} $ be the characteristic polynomials of $A(\Phi_{1}) $ and $ A(\Phi_{2}) $, respectively. Therefore, $ a_{i}=b_{i} $ for all $ i$. From Theorem \[Theo4\], $$a_i=\sum_{H\in\mathcal{H}_i(G)}(-1)^{p(H)}2^{c(H)}\prod_{C\in \mathcal{C}(H)} \Re (\varphi_{1}(C)), b_i=\sum_{H\in\mathcal{H}_i(G)}(-1)^{p(H)}2^{c(H)}\prod_{C\in \mathcal{C}(H)} \Re(\varphi_{2}(C)) .$$ Let $ Com{(m_{1}, m_{2},\dots,m_{k})} $ denote the collection of all elementary subgraphs $ H $ with $ k $ components such that the number of vertices in each component is $ m_{1}, m_{2}, \dots , m_{k}$, respectively, where $ m_{1} \geq m_{2} \geq \dots \geq m_{k} $. Let $ m=m_{1}+m_{2}+ \dots+m_{k} $ and $ S_{{(m_{1}, m_{2},\dots,m_{k})}}(\varphi_{1}) $ be the sum of the terms in the expression of $ a_{m} $ associated with the elementary subgraphs in $ Com{(m_{1}, m_{2},\dots,m_{k})} $. It is clear that $ a_{1}=b_{1} $, $ a_{2}=b_{2} $ for any $ \varphi_{1} $ and $ \varphi_{2} $. $ a_{3}=b_{3} $ gives us that $ \Re(\varphi_{1}(C))=\Re(\varphi_{2}(C)),$ where $ C $ denotes a cycle on three vertices if exists. By induction, we may assume that $ a_{j}=b_{j} $ for each $ j<m $ which implies that $ \Re(\varphi_{1}(C_{j}))=\Re(\varphi_{2}(C_{j})) $ for $ j<m $, where $C_{j} $ is the unique cycle of length $ j $ if exists. Now consider $ a_{m}=b_{m} $, therefore we have $$\label{eq1}
     S_{(m)}(\varphi_{1})+S_{(m-2,2)}(\varphi_{1})+S_{(m-3,3)} (\varphi_{1})+\cdots + S_{(m_{1}, m_{2},\dots,m_{k})}(\varphi_{1})+ \cdots $$
     $$ = S_{(m)}(\varphi_{2})+S_{(m-2, 2)}(\varphi_{2})+S_{(m-3,3)}(\varphi_{2})+\dots + S_{(m_{1}, m_{2},\dots,m_{k})}(\varphi_{2})+ \cdots$$ From the above expression, if a representing term $ S_{(m_{1}, m_{2},\dots,m_{k})}(\varphi_{1}) $ is nonzero, then we have\

$ S_{(m_{1}, m_{2},\dots,m_{k})}(\varphi_{1}) = (-1)^{k}2^{l}\Re( \varphi_{1}(C_{m_{1}}))\Re(\varphi_{1}(C_{m_{2}})) \cdots \Re(\varphi_{1}(C_{m_{l}}))$, $m_{l}>2$ and $ m_{l+1}=\dots = m_{k}=2 $.\

Therefore, except the first terms of both sides of equation (\[eq1\]), for any other terms,$ S_{(m_{1}, m_{2},\dots,m_{k})}(\varphi_{1})= S_{(m_{1}, m_{2},\dots,m_{k})}(\varphi_{2})$ due to $ m_{1} < m $. Hence from equation (\[eq1\]), we have $ S_{(m)}(\varphi_{1})=S_{(m)}(\varphi_{2})$. That is $ \Re(\varphi_{1}(C_{m}))=\Re(\varphi_{2}(C_{m})) $. Thus, by induction hypothesis, $ \Re(\varphi_{1}(C))=\Re(\varphi_{2}(C)) $ for all cycles $ C $ in $ G $. Converse is easy to verify.

$\mathbb{T}$-gains with nonnegative real part
=============================================

For any matrix $ A \in \mathbb{T}_{G} $, we have $ A=B+iC $, where $B$ and $C$ are real matrices. The real part of the matrix $ A =B+iC\in \mathbb{T}_{G} $ is denoted by $\Re(A)$, and is defined to be $B$. For a real matrix $M$, we write $ M \geq 0$, if all the entries of $M$ are nonnegative.

In this section, we construct a class of connected graphs, $ \mathfrak{F}^{'} $ such that for $ G \in \mathfrak{F}^{'} $, every $ \mathcal{A}_{T}(r) \subseteq \mathcal{T}_{G} $ contains at least one $ \Phi $ with $ \Re(A( \Phi )) \geq 0 $. For these class of graphs, in section $5$, we extend Theorem \[Theo.6\], which gives a bound for the spectral radius of the $\mathbb{T}$-gain adjacency matrices.

A connected graph $ G $ on $ n $ vertices and $ m $ edges with normal spanning tree $ T $ is said to have the property *gain nonnegative real part(GNRP)*, if for each $ r \in \mathbb{I}^{m-n+1} $ there exist a $ \Phi \in \mathcal{A}_{T}(r) $ such that $ \Re(A( \Phi )) \geq 0 $. The collection of all connected graphs on $ n $ vertices with the property GNRP is denoted by $ \mathcal{P}_{n} $ or simply by $ \mathcal{P} $ if there is no ambiguity.

First we show that the set of all connected graphs with vertex disjoint cycles is a sub collection of $ \mathcal{P} $. Let $ G $ be a connected graph with $ V(G)=\{v_{1}, v_{2}, \dots, v_{n} \} $ and $ m $ edges such that no two cycles have common vertex. Let $ T $ be a normal spanning tree of $ G $ with root vertex $ v_{r} $. Let $ \overrightarrow{C_{1}(T)}, \overrightarrow{C_{2}(T)}, \dots, \overrightarrow{C_{m-n+1}(T)}  $ be the directed fundamental cycles in $ \overrightarrow{G}_{T} $ on $ n_{1}, n_{2}, \dots, n_{(m-n+1)} $ vertices, respectively. For any cycle $ \overrightarrow{C_{k}(T)} $, consider $ u_{k}(j),\break j=1,2,\dots,n_{k}$ be its vertices and they are arranged in the ascending order with respect to tree ordering. That is, $ u_{k}(1)< u_{k}(2)\dots < u_{k}(n_{k}) $. The vertex $ u_{k}(1) $ is called the *leading vertex* of $ \overrightarrow{C_{k}(T)} $. The cycles corresponding to the leading vertices which are minimal in the set of all leading vertices is known as *leading cycles*. That is, if $ y_{1}, y_{2}, \dots, y_{s} $ are the pendant vertices of $ T $, then for each $ y_{j} $, there is at most one cycle, say $ \overrightarrow{C_{j}(T)} $ which intersects the path $  v_{r}Ty_{j} $ and nearest to the root $ v_{r} $, is called a *leading cycle* in $ \overrightarrow{G}_{T} $.

For any $ \Phi \in \mathcal{T}_{G}$, $ \varphi(\overrightarrow{v_{r}Tv_{y}}) $ is the gain of the directed path $ \overrightarrow{v_{r}Tv_{y}} $ in $ \Phi $, where $ v_{y} \in V(G) $. If $ \varphi(\overrightarrow{C_{k}(T)}) =e^{i\theta_{k}}$, for some $ \theta_{k} \in (-\pi , \pi] $, then the *average edge gain* of $ \overrightarrow{C_{k}(T)} $ is defined as $ e^ {i \psi_{k}}:=e^{i \frac{\theta_{k}}{n_{k}}} $ for each $ k=1, 2, \dots, (m-n+1) $. Therefore, it is clear that each $ \psi_{k} \in [ -\frac{\pi}{2}, \frac{\pi}{2}] $. See Figure $ \ref{fig5} $ for the illustration of the above definitions.

\

In the next theorem, we shall prove that the collection of all connected graphs with vertex disjoint cycles is a sub collection of $ \mathcal{P} $.

\[Theo4.1\] Let $ G $ be a connected graph on $ n $ vertices and $m$ edges. If all the cycles of $G$ are vertex disjoint, then $ G \in \mathcal{P}_{n} $.

Let $ T $ be a normal spanning tree of $ G $ with root vertex $ v_{r} $, and $ y_{1}, y_{2}, \dots , y_{s} $ be the pendent vertices in the tree $ T $. Let $ \overrightarrow{C_{1}(T)}, \overrightarrow{C_{2}(T)}, \dots, \overrightarrow{C_{m-n+1}(T)}  $ be the directed fundamental cycles in $ \overrightarrow{G}_{T} $ on $ n_{1}, n_{2}, \dots, n_{(m-n+1)} $ vertices, respectively. Without loss of generality, assume that first $ k $ of them are leading cycles. Let $ \varphi $ be any $ \mathbb{T} $-gain on $ G $. Then $ \Phi \in \mathcal{A}_{T}(w) $, for some $ w \in \mathbb{I}^{m-n+1} $. Let $ w=w^{'} $ ($\mod$ $ 2\pi $), where each entries of $ w^{'} $ is in $ (-\pi, \pi] $. Now, let us construct a $ \mathbb{T} $-gain graph on $ G $ which is switching equivalent to $\Phi$, and real part of whose gain adjacency matrix is nonnegative.

Let us consider the tree ordering of $V(G)$ with respect to the rooted tree $ T $. For any cycle $ \overrightarrow{C_{g}(T)} $, consider $ u_{g}(d), d=1,2,\dots,n_{g}$ be its vertices and they are arranged in the following ascending order with respect to the tree ordering. That is, $ u_{g}(1)< u_{g}(2)\dots < u_{g}(n_{g}) $. For each $ v_{z} \in V(G) $, assign weight on $ v_{z}$, denoted by $ W(v_{z}) $, as follows: If $ v_{z} \leq u_{j}(1) $, for $ j=1, 2, \dots, k $, then assign $ W(v_{z})=1 $. For each leading cycle $ \overrightarrow{C_{j}(T)}$, assign weight for each vertex in this cycle, recursively as follows: $$\label{eq.3}
   W(u_{j}(l))=e^{i(l-1)\psi_{j}}.W(u_{j}(1)), \text{ for  } l=2, \dots ,n_{j},$$

where $e^{ i \psi_{j}} $ is the average edge gain of $ \overrightarrow{C_{j}(T)} $. Consider the ordering of directed fundamental cycles corresponding to the ordering of their associated leading vertices. Now, recursively assign the weights for the vertices of nonleading cycles according to the ascending order. Let $ \overrightarrow{C_{q}(T)}$ be a cycle which is next to some leading cycle $ \overrightarrow{C_{j}(T)}$ . Then $ u_{q}(1) $ must be connected with some vertex $ u_{j}(s) $, for some $s$. The weight of any vertex in the path $ \overrightarrow{u_{j}(s)Tu_{q}(1)} $ is $W( u_{j}(s)) $, and for other vertices of $ \overrightarrow{C_{q}(T)}$ assign the weights using the equation (\[eq.3\]). For a pendent vertex $ y_{j} $, if the vertex $ u_{p}(t) $ is such that $ \overrightarrow{u_{p}(t)Ty_{j}} $ does not intersect any cycle, then the weight of any vertex on the path $ \overrightarrow{u_{p}(t)Ty_{j}} $ is given by $ W(u_{p}(t)) $. If the path $ \overrightarrow{v_{r}Ty_{j}} $ does not intersect any cycle, then assign the weight $1$ to all the vertices in this path. Let us define a switching function $ \zeta $ on $ V(G) $ as follows: For each $v_{z} \in V(G)$,

$\zeta(v_{z})=W(v_{z})\varphi(\overrightarrow{v_{r}Tv_{z}})^{-1}.$

Let $\varphi^{\zeta}$ denote the gain function induced by $\zeta$. Let $ \overrightarrow{e_{xy}} $ be an edge in $ \overrightarrow{G}_{T}$ which is not in any cycle. If the path $ \overrightarrow{v_{r}Tv_{x}} $ does not intersect any cycle, then it is easy to see $   \varphi^{\zeta}(\overrightarrow{e_{xy}}) =1$. Let the path $ \overrightarrow{v_{r}Tv_{x}} $ intersect some of the cycle. Assume that the vertex $ u_{j}(l)$ in the path $ \overrightarrow{v_{r}Tv_{x}} $ is the largest element in the path which is in some cycle $ \overrightarrow{C_{j}(T)} $. That is, $\overrightarrow{C_{j}(T)} $ is the nearest cycle to $v_{x}$ which intersects the path $\overrightarrow{v_{r}Tv_{x}}$, and $u_{j}(l)$ is the vertex in the cycle $\overrightarrow{C_{j}(T)} $ which is closest to $v_{x}$. Then,

$$\begin{aligned}
    \varphi^{\zeta}(\overrightarrow{e_{xy}}) &=\zeta^{-1}(v_{x})\varphi(\overrightarrow{e_{xy}})\zeta(v_{y}) \\
    &=W(v_{x})^{-1}\varphi(\overrightarrow{v_{r}Tv_{x}}) \varphi(\overrightarrow{e_{xy}})W(v_{y}) \varphi(\overrightarrow{v_{r}Tv_{y}})^{-1} \\
     &=\{e^{i(l-1)\psi_{j}}W(u_{j}(1))\}^{-1}\varphi(\overrightarrow{v_{r}Tv_{x}})\varphi(\overrightarrow{e_{xy}})e^{i(l-1)\psi_{j}}W(u_{j}(1))\varphi(\overrightarrow{v_{r}Tv_{y}})^{-1} \\
    &=1.\end{aligned}$$

Let $ \overrightarrow{e_{xy}} $ be an edge in some cycle $ \overrightarrow{C_{j}(T)} $, and assume that $ v_{x}=u_{j}(h), v_{y}=u_{j}(h+1)$. Then we have,

$$\begin{aligned}
    \varphi^{\zeta}(\overrightarrow{e_{xy}})&= \zeta^{-1}(v_{x})\varphi(\overrightarrow{e_{xy}})\zeta(v_{y})\\
    &= \zeta^{-1}(u_{j}(h))\varphi(\overrightarrow{e_{xy}})\zeta(u_{j}(h+1))\\
     &=W(u_{j}(h))^{-1}\varphi(\overrightarrow{v_{r}Tu_{j}(h)})\varphi(\overrightarrow{e_{xy}})W(u_{j}(h+1))\varphi(\overrightarrow{v_{r}Tu_{j}(h+1)})^{-1}\\
    &= e^{i\psi_{j}}.\end{aligned}$$

Since $ \psi_{j} \in [-\frac{\pi}{2}, \frac{\pi}{2}] $, therefore the real part of $ \varphi^{\zeta}(\overrightarrow{e_{xy}}) $ is nonnegative. Thus $ \varphi^{\zeta} \in \mathcal{A}_{T}(w
    ) $ and $ \Re(A(\Phi^{\zeta})) \geq 0 $.

Let us illustrate the above theorem in the following example.

Let us consider a $ \mathbb{T} $-gain adjacency matrix, $ A(\Phi) \in \mathbb{T}_{G} $ such that\
$$A(\Phi)=
     \begin{bmatrix}
     0 & -1 & \frac{1-i}{\sqrt{2}} & 0 & 0 & 0 & 0 \\
     -1 & 0 & i & 0 & 0&0 & 0 \\
     \frac{1+i}{\sqrt{2}}& -i & 0 & i & 0 & 0 & 0 \\
     0 & 0 & -i & 0 & 1 & 0 & \frac{-1-i}{\sqrt{2}} \\
     0 & 0 & 0 & 1 & 0 & \frac{-1+i \sqrt{3}}{2} & 0 \\
     0 & 0 & 0 & 0 & \frac{-1-i \sqrt{3}}{2} & 0 & -1 \\
     0 & 0 & 0 & \frac{-1+i}{\sqrt{2}} & 0 & -1 & 0
     \end{bmatrix},$$ where $ G $ is a connected graph with vertex disjoint cycles given in Figure \[fig6\].

\

Using the above technique, we can find $ \varphi^{\zeta} $, such that\
$$A(\Phi ^{\zeta})=
     \begin{bmatrix}
     0 & \overline{\alpha} & \alpha & 0 & 0 & 0 & 0 \\
     \alpha  & 0 & \overline{\alpha}  & 0 & 0&0 & 0 \\
     \overline{\alpha} & \alpha  & 0 & 1 & 0 & 0 & 0 \\
     0 & 0 & 1 & 0 & \beta & 0 & \overline{\beta} \\
     0 & 0 & 0 & \overline{\beta} & 0 & \beta & 0 \\
     0 & 0 & 0 & 0 & \overline{\beta} & 0 & \beta \\
     0 & 0 & 0 & \beta & 0 & \overline{\beta}& 0
     \end{bmatrix}$$ where $ \alpha = \frac{(1+\sqrt{3})+i(\sqrt{3}-1)}{2\sqrt{2}}, \beta=\frac{1}{2}(\sqrt{2+\sqrt{2-\sqrt{3}}} +i \sqrt{2-\sqrt{2-\sqrt{3}}}) $ and $\Re(\alpha), \Re(\beta) \geq 0 $.

Now we are interested in identifying more classes of graphs belong to the class $\mathcal{P}_n$. Let $ C_{1}(T), C_{2}(T), \dots, C_{k}(T) $ be the collection of all fundamental cycles of a connected graph $ G $ associated with a normal spanning tree $ T $. For simplicity, we use $ C_{k} $ instead of $ C_{k}(T) $. Consider a labeling graph $ G $. For $ j=1,2 $, let $ G_{j}=( V(G_{j}), E(G_{j}))$ be two subgraphs of $ G $. Then the sum of these subgraphs $G_1$ and $G_2$ is denoted by $G_1 + G_2$, and is defined as the subgraph which consists of all the edges in $G_1$ or $G_2$ or both [@ore].

Let $G$ be a connected graph and $T$ be a normal spanning tree of $G$. Let $ C_{1}, C_{2}, \dots, C_{s} $ be a collection of fundamental cycles such that intersection of the subgraph $C_1 + \cdots + C_s$ in $G$ and the normal spanning tree $T$ is a subtree. A subgraph which is the sum of a maximal collection of fundamental cycles of $G$ with respect to the above property is called a *fundamental subgraph* of a graph $ G $ with respect to $ T $. It is denoted by $ \mathfrak{B}(T;C_{1},\dots, C_{s}) $.

Suppose a connected graph $ G $ has $ m $ fundamental subgraphs with respect to some normal spanning tree $ T $, say $\mathfrak{B}_{1}, \mathfrak{B}_2,\dots , \mathfrak{B}_{m} $. Then all of them are vertex disjoint. If we contract each fundamental subgraph of $G$ to a vertex, then the resultant graph is a tree. Let $ \mathfrak{B} $ be a fundamental subgraph of $ G $ containing the only fundamental cycles namely, $ C_{1}, C_{2}, \dots, C_{s} $. Then it is always possible to arrange the fundamental cycles in a sequence, say $ (C_{k_1}, C_{k_2}, \dots, C_{k_s}) $ such that $|E(C_{k_i} \setminus (C_{k_1}+ C_{k_2}+ \cdots + C_{k_{i-1}}))| \geq 1$ for $ i =2, 3, \dots, s$. Therefore, whenever we write $ \mathfrak{B}(T;C_{1}, C_{2}, \dots, C_{s}) $, without loss of generality, we assume that the arrangement of the fundamental cycles $(C_{1}, C_{2}, \dots, C_{s})$ has the above property.

Let $G$ be a connected graph and $T$ be a normal spanning tree of $G$. A fundamental subgraph $ \mathfrak{B}(T;C_{1},\dots, C_{s}) $ of $ G $ associated with $ T $ is said to have *distinguished edge property(DEP)* if we can arrange the fundamental cycles in a sequence, say $(C_{k_1},\dots, C_{k_s})$, such that $|E(C_{k_i} \setminus (C_{k_1}+ C_{k_2}+ \cdots + C_{k_{i-1}}))|>1$, for all $ i=2, 3, \dots, s $. A connected graph $G$ is said to have *distinguished edge property(DEP)*, if every fundamental subgraph of $G$ with respect to $ T $ has distinguished edge property(DEP). Let $ \mathfrak{F}$ denote the collection of all connected graphs having DEP.

If $ \mathfrak{B}(T;C_{1}, C_{2}, \dots, C_{s}) $ is a fundamental subgraph of a graph $ G \in \mathfrak{F} $, then, without loss of generality we assume that $( C_{1}, C_{2},\dots, C_{s})$ is the sequence of fundamental cycles satisfy DEP. Any connected graph $ G $ with vertex disjoint cycles has the property DEP. We say a $ \mathbb{T} $-gain graph $ \Phi=(G, \varphi) $ is said to have the property DEP if its underlying graph $ G $ has DEP.

Let us illustrate the above definitions in the following example.

Let $ G $ and $ \overrightarrow{G}_{T} $ be given as in Figure \[fig7\] and Figure \[figu8.1\], respectively. A normal spanning tree $ T $ of $ G $ is denoted by the dotted lines in Figure \[fig7\]. Then there are two fundamental subgraphs of $ G $ with respect to $ T $. One of them is $ \mathfrak{B}_{1}(T;C_{1}, C_{2}, C_{3}, C_{4}, C_{5}) $, where $C_{1}, C_{2}, C_{3}, C_{4}, C_{5}$ are defined as follows: $ C_{1}\equiv v_{3}e_{34}v_{4}e_{45}v_{5}e_{53}v_{3}$, $C_{2}\equiv v_{2}e_{23}v_{3}e_{34}v_{4}e_{45}v_{5}e_{52}v_{2}$, $C_{3}\equiv v_{1}e_{12}v_{2}e_{23}v_{3}e_{31}v_{1}$, $  C_{4}\equiv v_{2}e_{23}v_{3}e_{3(14)}v_{14}e_{(14)2}v_{2},$ $ C_{5}\equiv v_{2}e_{23}v_{3}e_{3(14)}v_{14}e_{(14)(15)}v_{15}e_{(15)2}v_{2}$.

![Suitably oriented graph of $ G $ []{data-label="figu8.1"}](fig7){width="\linewidth"}

![Suitably oriented graph of $ G $ []{data-label="figu8.1"}](fig8){width="\linewidth"}

\[Theo.4.2\] If a connected graph $ G \in \mathfrak{F} $, then $ G \in \mathcal{P} $. That is, $ \mathfrak{F} \subset \mathcal{P} $.

Let $\mathfrak{B}_{1}, \mathfrak{B}_2, \dots, \mathfrak{B}_{k} $ be the fundamental subgraphs of a connected graph $ G $ associated with a normal spanning tree $ T $. Let $ C_{1}, C_{2}, \dots, C_{m-n+1} $ be the fundamental cycles of $ G $ with respect to $ T $ and $ \overrightarrow{C_{1}(T)}, \overrightarrow{C_{2}(T)}, \dots, \overrightarrow{C_{m-n+1}(T)}$ be the associated directed fundamental cycles in $ \overrightarrow{G_{T}} $. Without loss of generality, let us assume that $ \mathfrak{B}_{1}=\mathfrak{B}_{1}(C_{1}, C_{2}, \dots, C_{s}) $. Let $ \mathcal{A}_{T}(r) $ be an arbitrary equivalence class of $ \mathcal{T}_{G} $, where $ r=(r_{1}, r_{2}, \dots, r_{m-n+1}) \in \mathbb{I}^{m-n+1}$. Let $ {r_{j}}^{'}=r_{j}$ ($\mod$ $ 2\pi $), where $ {r_{j}}^{'}\in \left(-\pi, \pi \right] $ for $ j=1,2, \dots, (m-n+1) $. In rest of the proof, we always do scaling in the region $\left(-\pi, \pi \right] $. Let $ \Phi \in  \mathcal{A}_{T}(r) $, then $ \varphi(\overrightarrow{C_{j}(T)})=e^{i{r_{j}}^{'}} $ for $ j=1,2, \dots, (m-n+1) $. Let us construct $ \Phi^{'}=(G, \varphi^{'}) $ from $ \Phi $ such that $ \Phi^{'}\in\mathcal{A}_{T}(r)$ with $ \Re(A(\Phi^{'})) \geq 0 $. Now, define the gain function $\varphi^{'}$ recursively as follows: Let $ \theta_{1}=\frac{{r_{1}}^{'}}{|E(C_{1})|}$, where $ |E(C_{1})|$ is the number of edges of $ C_{1} $. For each edge $\overrightarrow{e}$ of $ \overrightarrow{C_{1}(T)} $, define $ \varphi^{'} (\overrightarrow{e})$ to be the average edge gain of the directed cycle, $\overrightarrow{C_{1}(T)}$. That is, $ \varphi^{'} (\overrightarrow{e})=e^{i\theta_{1}}$. Then for each edge of $ \overrightarrow{C_{2}(T)} \setminus \overrightarrow{C_{1}(T)} $, assign the gain as follows: Let $ |E(C_{1}\cap E(C_{2} )| =l_{1}$ and $ |E(C_{2})\setminus E(C_{1})|=k_{1} $. Consider $ \gamma_{1}= (r_{2}-\theta_{1}l_{1}) (\mod 2\pi)$ and scale it to the interval $ \left(-\pi, \pi \right] $. Then for each edge of $ \overrightarrow{C_{2}(T)} \setminus \overrightarrow{C_{1}(T)} $, assign the gain $ e^{i \frac{\gamma_{1}}{k_{1}}} $. As $G \in \mathfrak{F} $, we have $k_{1} \geq 2$. Thus the real part of $ e^{i \frac{\gamma_{1}}{k_{1}}} $ is nonnegative. Let $ \theta_{2}=\frac{\gamma_{1}}{k_{1}} $. At the end of the process, we get $ e^{ i\theta_{1}}, e^{ i\theta_{2}}, \dots, e^{ i\theta_{s}}$ as the only possible resultant gains of the oriented edges of $\mathfrak{B}_{1} $, where $ \theta_{t}\in [-\frac{\pi}{2}, \frac{\pi}{2}]$ for $ t=1,2, \dots, s $. Thus the real part of gain of each oriented edge of $ \mathfrak{B}_{1} $ is nonnegative. Continue the same process for other fundamental subgraphs. Finally, assign gain $ 1 $ for edges which are not in any fundamental subgraphs. Since under this procedure, gains of $ \overrightarrow{C_{1}(T)}, \overrightarrow{C_{2}(T)}, \dots , \overrightarrow{C_{m-n+1}(T)}$ remain unchanged, so $ \Phi^{'}\in \mathcal{A}_{T}(r) $, and by the construction $ \Re(A(\Phi^{'})) \geq 0 $.

Next theorem shows that not all graphs in $\mathcal{P}$ are in $ \mathfrak{F} $.

\[Theo.4.3.\] Consider $ K_{4} $, the complete graph on $4$ vertices. It is easy to see that $ k_{4} \notin \mathfrak{F} $. However, $ K_{4} \in \mathcal{P} $.

Let us consider a normal spanning tree of $ K_{4} $ by $ T\equiv v_{1}e_{12}v_{2}e_{23}v_{3}e_{34}v_{4} $ with root vertex $ v_{1} $ (see Figure \[fig9\]). Then the directed fundamental cycles of $ K_{4} $ with respect to $ T $ are $\overrightarrow{ C_{1}(T)}\equiv v_{1}\overrightarrow{e_{12}}v_{2}\overrightarrow{e_{23}}v_{3}\overrightarrow{e_{34}}v_{4}\overrightarrow{e_{41}}v_{1}$, $ \overrightarrow{C_{2}(T)}\equiv v_{2}\overrightarrow{e_{23}}v_{3}\overrightarrow{e_{34}}v_{4}\overrightarrow{e_{42}} v_{2}$ and $ \overrightarrow{C_{3}(T)}\equiv v_{1}\overrightarrow{e_{12}}v_{2}\overrightarrow{e_{23}}v_{3}\overrightarrow{e_{31}}v_{1}$. Let $ r=(\theta_{1}, \theta_{2}, \theta_{3})\in \mathbb{I}^{3} $ and $ \Phi \in \mathcal{A}_{T}(r) $. Then $ \varphi(\overrightarrow{C_{m}(T)})=e^{i \theta_{m}} $, $ m=1,2,3, $ respectively.

![ Digraph $ X $ and suitably oriented graph of $ K_{4} $ []{data-label="fig9"}](fig9)

Now construct a switching function $ \zeta $ such that $ \varphi^ \zeta (\overrightarrow{e_{12}}) = e^{ix_{1}}, \varphi^ \zeta (\overrightarrow{e_{23}}) = e^{ix_{2}},\varphi^ \zeta(\overrightarrow{e_{34}}) = e^{ix_{3}}, \varphi^ \zeta(\overrightarrow{e_{41}}) = e^{ix_{4}}, \varphi^ \zeta(\overrightarrow{e_{42}}) = e^{ix_{5}},$ and $\varphi^ \zeta(\overrightarrow{e_{31}}) =  e^{ix_{6}}, $ with $ x_{i} \in [-\frac{\pi}{2}, \frac{\pi}{2}] $, for all$i=1,2, \dots, 6. $ Let us consider the system of linear equations:

$ x_{1}+x_{2}+x_{3}+x_{4}=\theta_{1} $; $  x_{2}+x_{3}+x_{5}=\theta_{2} $; $ x_{1}+x_{2}+x_{6}=\theta_{3}.$

To make the computations easier, take $ x_{p}=y_{p}-\frac{\pi}{2} $, for $ p=1,2, ..,6 $. Then,

$ y_{1}+y_{2}+y_{3}+y_{4}=\theta^{'}_{1} $; $  y_{2}+y_{3}+y_{5}=\theta^{'}_{2} $; $ y_{1}+y_{2}+y_{6}=\theta^{'}_{3} $,

the left hand and right hand side values are taken up to modulo $2 \pi$. Now, we construct $y_i$’s depending up on the values of $\theta^{'}_{i}$’s.

[**Case-I:**]{} If $ \theta^{'}_{1}, \theta^{'}_{2}$, $ \theta^{'}_{3} \in [0,\pi] $, then take $ y_{1}=y_{2}=y_{3}=0 $ and $y_{4}= \theta^{'}_{1}, y_{5}=\theta^{'}_{2}, y_{6}=\theta^{'}_{3} $.

[**Case-II:**]{} Exactly two of the $\theta^{'}_{i}$’s are less than or equal to $ \pi $.

If $ \theta^{'}_{1},\theta^{'}_{2} \in[ 0,\pi ] $ and $ \theta^{'}_{3}> \pi $, then take $ y_{1}=0, y_{2}=\pi , y_{3}=\pi  , y_{4}=\theta^{'}_{1} , y_{5}=\theta^{'}_{2} , y_{6}=\theta^{'}_{3}-\pi$.

If $ \theta^{'}_{1},\theta^{'}_{3} \in[ 0,\pi ] $ and $ \theta^{'}_{2}> \pi $, then take $ y_{1}= \pi, y_{2}= \pi , y_{3}= 0, y_{4}= \theta^{'}_{1}, y_{5}=\theta^{'}_{2}-\pi , y_{6}=\theta^{'}_{3}.$

If $ \theta^{'}_{2},\theta^{'}_{3} \in[ 0,\pi ] $ and $ \theta^{'}_{1}> \pi $, then take $ y_{1}=y_{2}=y_{3}=\pi , y_{4}=\theta^{'}_{1}-\pi , y_{5}=\theta^{'}_{2}, y_{6}= \theta^{'}_{3}.$

[**Case-III:**]{} Exactly one of $\theta^{'}_{i}$’s is less than or equal to $ \pi $.

If $ \theta^{'}_{1} \in[ 0,\pi ] $ and $\theta^{'}_{3}, \theta^{'}_{2}> \pi $, then take $ y_{1}= y_{3}= \pi, y_{2}=0 , y_{4}=\theta^{'}_{1} , y_{5}=\theta^{'}_{2}-\pi, y_{6}=\theta^{'}_{3}-\pi.$

If $ \theta^{'}_{2} \in[ 0,\pi ] $ and $ ,\theta^{'}_{1}, \theta^{'}_{3}> \pi $, then take $ y_{2}= y_{3}=0, y_{1}= \pi ,y_{4}=\theta^{'}_{1}-\pi  , y_{5}= \theta^{'}_{2}, y_{6}=\theta^{'}_{3}-\pi.$

If $ \theta^{'}_{3} \in[ 0,\pi ] $ and $ \theta^{'}_{1}, \theta^{'}_{2}> \pi $, then take $ y_{1}=y_{2}=0 , y_{3}=\pi, y_{4}=\theta^{'}_{1}-\pi , y_{5}=\theta^{'}_{2}-\pi , y_{6}= \theta^{'}_{3}.$

[**Case-IV:**]{} All $\theta^{'}_{i}$’s are strictly larger than $ \pi $. That is, $\theta^{'}_{1},\theta^{'}_{2}, \theta^{'}_{3} > \pi    $, then take $ y_{1}=0, y_{2}=\pi , y_{3}=0 , y_{4}=\theta^{'}_{1}-\pi , y_{5}=\theta^{'}_{2}-\pi , y_{6}= \theta^{'}_{3}-\pi $.

From the above assignment, $y_i \in [0, \pi]$, for $ i=1,2,\dots,6 $. Therefore $ x_{i} \in [-\frac{\pi}{2}, \frac{\pi}{2}] $, for all $i=1,2, \dots, 6$. Thus $ \Re(A(\Phi^{\zeta})) \geq 0 $. Since $ r $ is arbitrary, so $ K_{4} \in \mathcal{P} $.

\[D24, [@Gross]\] A *subdivision of an edge* $ e_{xy} $ in a graph $ G $ is obtained by replacing the edge with a new vertex $ v_{z} $ and two new edges, $ e_{xz} $ and $ e_{zy} $. A graph $ H $ is called a *$ G $ subdivision* if $ H $ can be constructed from $ G $ by a sequence of edge subdivisions.

It is clear that the graphs $ K_{4} $ and $ K_{4} $ subdivisions have exactly three fundamental cycles with respect to any normal spanning tree. For simplicity, we denote $ {K_{4}}^{'} $ as either $ K_{4} $ or $ K_{4} $ subdivision.

Let $ G $ be a connected graph with normal spanning tree $ T $. Let $ \mathfrak{B} $ be a fundamental subgraph of $ G $ associated with $ T $, and $ S $ be the set of all fundamental cycles in $ \mathfrak{B} $ with respect to $ T $. Then the fundamental subgraph $ \mathfrak{B} $ is called $ {K_{4}}^{'}$-*initiated fundamental subgraph* of $ G $ associated with $ T $, if $ S $ contains three cycles say, $ {C_{1}}^{'}, {C_{2}}^{'}$ and $ {C_{3}}^{'} $ such that $ ({C_{1}}^{'}+{C_{2}}^{'}+{C_{3}}^{'})={K_{4}}^{'} $, and is denoted by $ \mathfrak{B}(T;{K_{4}}^{'}, C_{1}, \dots, C_{s})$, and is defined by ${K_{4}}^{'}+ C_{1}+ \dots+ C_{s}$. Here $ C_{1}, C_{2}, \dots, C_{s} $ are the fundamental cycles other than $ {C_{1}}^{'}, {C_{2}}^{'}$ and $ {C_{3}}^{'} $ in $ S $.

A fundamental subgraph which is not $ K_{4}^{'} $-initiated fundamental subgraph is known as *non-$ K_{4}^{'} $-initiated fundamental subgraph*. Therefore any fundamental subgraph is one of these two types.

Let $G$ be a connected graph and $T$ be a normal spanning tree of $G$. A $ K_{4}^{'} $-*initiated fundamental subgraph* $ \mathfrak{B}(T;{K_{4}}^{'}, C_{1}, \dots, C_{s})$ of $ G $ associated with $ T $ is said to have *distinguished edge property(DEP)* if there is a sequence of fundamental cycles other than $K_{4}^{'}$ , say $(C_{k_1},\dots, C_{k_s})$, such that $|E(C_{k_i} \setminus (C_{k_0} + C_{k_1} + \cdots + C_{k_{i-1}})|>1$, for all $ i=1, 2, \dots, s $, where $ C_{0}={K_{4}}^{'} $.

A connected graph $ G $ is said to have *extended distinguished edge property (EDEP)* if its all $ K_{4}^{'}$-initiated fundamental subgraphs and non-$K_{4}^{'} $-initiated fundamental subgraphs associated with a normal spanning tree $ T $ have distinguished edge property. Collection of all connected graphs with the property EDEP is denoted by $ \mathfrak{F^{'}} $.

It is clear that $ \mathfrak{F} \subseteq  \mathfrak{F^{'}}  $. Next example shows that the inclusion is proper.

Consider the graph given in Figure \[fig11.0\]. Let $ T $ be a normal spanning tree of $ G $ denoted by the undirected dotted lines in Figure \[fig11.0\]. If we choose $ K_{4} $, the subgraph induced by the vertices $ \{v_{3}, v_{4}, v_{5}, v_{6} \} $ and the other fundamental cycles, $ C_{1} \equiv v_{2}e_{23}v_{3}e_{34}v_{4}e_{45}v_{5}e_{52}v_{2}, C_{2}\equiv v_{1}e_{12}v_{2}e_{23}v_{3}e_{31}v_{1}, C_{3} \equiv v_{2}e_{23}v_{3}e_{38}v_{8}e_{82}v_{2} $ and $ C_{4} \equiv v_{2}e_{23}v_{3}e_{38} v_{8}e_{89}v_{9}e_{92}v_{2} $ then associated with the sequence $ (C_{1}, C_{2}, C_{3}, C_{4}) $, $\mathfrak{B}(T; K_{4}, C_{1}, C_{2}, C_{3},C_{4})$ has to have DEP. And hence $ G \in \mathfrak{F^{'}}  $ but as $ K_{4} $ does not have DEP, so $ G \notin \mathfrak{F} $.

![ Simple graph $G \in \mathfrak{F^{'}}$ and its suitably oriented graph. []{data-label="fig11.0"}](fig11)

\[Theo.4.3\] $ \mathfrak{F}^{'} \subseteq \mathcal{P} $

Let $ G \in \mathfrak{F^{'}} $ be a connected graph and $ T $ be a normal spanning tree of $ G $. Let $ \mathcal{A}_{T}(r) $ be an equivalence class of $\mathcal{T}_{G} $. Let $ \Phi \in \mathcal{A}_{T}(r) $. Let us construct a switching function $ \zeta $ such that $ \Re(A(\Phi^{\zeta})) \geq 0 $. Since $ G \in \mathfrak{F^{'}}$, all the $ K_{4}^{'} $-initiated fundamental subgraphs and non-$ K_{4}^{'} $-initiated fundamental subgraphs have DEP with respect to $ T $. Let $ \mathfrak{B} = \mathfrak{B}( T;{K_{4}}^{'}, C_{1}, C_{2}, \dots, C_{s})$ be a $ K_{4}^{'} $-initiated fundamental subgraph. Then construct $ \zeta $ as follows: Assign gain of each edge in $ K_{4} $ subdivision as in Theorem \[Theo.4.3.\]. Then recursively, assign gains for other edges in $ C_{1} \setminus {K_{4}}^{'} $ as in the proof of Theorem \[Theo.4.2\]. Suppose $ \mathfrak{B} $ is a non-$ K_{4}^{'} $-initiated fundamental subgraph, then assign gains for edges of $ \mathfrak{B} $ as in Theorem \[Theo.4.2\]. In both cases, the resultant gain of each edge has nonnegative real part. That is, $ \Re(A(\Phi^{\zeta})) \geq 0 $. Since $ \mathcal{A}_{T}(r) $ is arbitrary, hence $ G \in \mathcal{P} $.

Find all the graphs in $ \mathcal{P}_{n} $.

 $k$-generalized Hermitian adjacency matrices and spectral radius  {#spec-bound}
==================================================================

In this section, we establish bounds for the spectral radius of $ \mathbb{T}$-gain adjacency matrices associated with connected graphs. For Hermitian adjacency matrices of digraphs , in [@Bojan Theorem 5.6] the authors established the following bounds for the spectral radius of the Hermitian adjacency matrices $ H(G) $ in terms of the largest eigenvalue $\lambda_{1}(H(G))$ of $H(G)$: $$\label{guo-mohar-ineq}
     \lambda_{1}(H(G))) \leq \rho(H(G)) \leq 3\lambda_{1}(H(G)) .$$ Since the Hermitian adjacency matrices are particular cases of $ \mathbb{T} $-gain adjacency matrices, it is natural to ask the following question: can we extend the bounds given in (\[guo-mohar-ineq\]) for the class of $ \mathbb{T}$-gain adjacency matrices? Two of the main results (Theorem \[Theo.5.1\] and Theorem \[Theo.5.2\]) of this section answer this question partially. In [@Bojan Theorem 5.1], the authors shown that the spectral radius of the Hermitian adjacency matrix of a digraph is bounded above by the maximum vertex degree of the underlying simple graph, and characterized the graphs for which the equality holds. In Theorem \[Theo.4.5\], we extend this result for $\mathbb{T}$-gains graphs. Next, we define the notion of $ k $-generalized Hermitian adjacency matrix of a digraph, which is an extension of Hermitian adjacency matrix of a digraph. Finally, in Theorem \[Theo5.5\], we characterize the structure of a digraph $ X $ such that the spectral radius of $k$-generalized Hermitian adjacency matrix of $ X $ is the largest vertex degree of its underlying graph, which gives an alternate proof of [@Bojan Theorem 5.1].

\[Theo.5.1\] Let $ G $ be any connected graph and $\varphi $ be a $ \mathbb{T} $-gain on $ G $ such that $ \varphi(\overrightarrow{E}(G)) \subseteq \{a+ib \in \mathbb{T}: a \geq 0 \} $. Then $ \lambda_{1}(A(\Phi))\leq \rho(A(\Phi))\leq3\lambda_{1}(A(\Phi))$.

Let $ \lambda_{1} \geq \lambda_{2} \geq \dots\geq \lambda_{n} $ be the eigenvalues of $A(\Phi)$. Let us split the adjacency matrix $ A(\Phi) = P+R $, where $ P = \Re(A( \Phi )) $, the real part of $A(\Phi)$, and $ R  = A(\Phi) - \Re(A( \Phi ))$. It is easy to see that the entries of the matrix $R$ are either $0$ or purely imaginary. For every $ x\in \mathbb{R}^{n}$, $x^{T}Rx=0 $. Thus, for $ x\in \mathbb{R}^{n}$, we get $ x^{*}A(\Phi) x =x^{T}P x $. Now, $$\begin{aligned}
    \lambda_{1}(P)& =\max_{x\in \mathbb{R}^{n}, ||x||=1}(x^{T}Px)\\
    &=\max_{x\in \mathbb{R}^{n}, ||x||=1}(x^{*}A(\Phi) x)\\
    &\leq \max_{x\in \mathbb{C}^{n}, ||z||=1}(z^{*}A(\Phi) z)\\
    &=\lambda_{1}(A(\Phi)).\end{aligned}$$

Suppose that $ \rho(A(\Phi))> 3\lambda_{1}(A(\Phi)) $. Then $ \lambda_{1}(P)\leq \lambda_{1}(A(\Phi))< \frac{1}{3} \rho(A(\Phi))$. Since $ \rho(A(\Phi))=-\lambda_{n}(A(\Phi)) =\Big|\min\limits_{z\in \mathbb{C}^{n},||z||=1}(z^{*}A(\Phi) z)\Big|=|w^{*}A(\Phi) w|$, for some $ w\in \mathbb{C}^{n}$ with $ ||w||=1$, we have $$\begin{aligned}
    \rho(A(\Phi))&=|w^{*}A(\Phi) w|\\
    &\leq |w^{*}P w|+|w^{*}R w|\\
    &\leq |w^{*}R w|+\rho(P) \\
    &\leq |w^{*}R w|+\lambda_{1}(P) ~~~~~~[\mbox{By Perron-Frobenius theorem}]\\
    &\leq |w^{*}R w|+\frac{1}{3}\rho(A(\Phi)).\end{aligned}$$ Thus $ |w^{*}R w|\geq\frac{2}{3}\rho(A(\Phi)).$ Now, the matrix $ R $ can be written as $ R=iL $, where $ L $ is a skew-symmetric matrix with real entries. Therefore, the eigenvalues of $ L $ are either zero or purely imaginary, and they occur in conjugate pair. Hence the eigenvalues of $ R = iL $ are symmetric about the origin. Since $ R$ is a Hermitian matrix, so by the Theorem \[Theo.5\], the numerical range of $ R $ is $ [-\rho(R),\rho(R)] $. Therefore $ w^{*}R w \in [-\rho(R),\rho(R)] $ and hence there exists $ y \in \mathbb{C}^{n}$ with $ || y||=1 $ such that $ y^{*}Ry  = |w^{*}Rw|$.\
Now, $$\begin{aligned}
    \lambda_{1}(A(\Phi))&=\max_{z\in \mathbb{C}^{n}, ||z||=1}(z^{*}A(\Phi) z),\\
    & \geq y^{*}A(\Phi) y,\\
    &= y^{*} P y+y^{*} R y,\\
    &\geq \frac{2}{3} \rho(A(\Phi))-\rho(P) \geq \frac{1}{3} \rho (A(\Phi)),\end{aligned}$$

which is absurd. Thus, we have $\lambda_{1}(A(\Phi))\leq \rho(A(\Phi))\leq3\lambda_{1}(A(\Phi)).$

In the next theorem, we impose a condition on the structure of the graph so that the bounds for the spectral radius given in (\[guo-mohar-ineq\]) hold true.

\[Theo.5.2\] Let $G$ be a connected graph on $n$ vertices and $m$ edges. If $ G \in \mathfrak{F^{'}} $, then for any $ \Phi \in \mathcal{T}_{G} $, we have $\lambda_{1}(A(\Phi))\leq \rho(A(\Phi))\leq3\lambda_{1}(A(\Phi))$.

Since $ G \in \mathfrak{F^{'}} $, then by Theorem \[Theo.4.3\], $ G \in \mathcal{P} $. Therefore, for any $ r \in \mathbb{I}^{m-n+1} $, there exists a $ \Phi \in \mathcal{A}_{T}(r) $ such that $ \Re(A(\Phi)) \geq 0 $. By Theorem \[Theo.5.1\], $ \lambda_{1}(A(\Phi))\leq \rho(A(\Phi))\leq 3\lambda_{1}(A(\Phi)) $, and hence for any $ \Psi \in \mathcal{A}_{T}(r) $, $ \lambda_{1}(A(\Psi))\leq \rho(A(\Psi))\leq 3\lambda_{1}(A(\Psi)) $. Since $ r $ is arbitrary, so for any $ \Phi \in \mathcal{T}_{G} $, $ \lambda_{1}(A(\Phi))\leq \rho(A(\Phi))\leq 3\lambda_{1}(A(\Phi)) $.

\[conj1\] Let $G$ be a connected graph and $ \Phi \in \mathcal{T}_{G} $. Then $\lambda_{1}(A(\Phi))\leq \rho(A(\Phi))\leq3\lambda_{1}(A(\Phi))$.

Next result characterizes the sharpness of the bound established in Theorem \[Theo.12\].

\[Theo.4.5\] Let $ \Phi $ be a $ \mathbb{T}$-gain graph on a connected graph $ G $ with maximum vertex degree $ \Delta $. Then $ \rho(A(\Phi))=\Delta $ if and only if $ G $ is $ \Delta $-regular and either $ \Phi $ or $ -\Phi $ is balanced.

Proof follows from Theorem \[Theo.9\] and Theorem \[Theo.11\].

Our next goal is to extend Theorem \[Theo.2.6\] for $\mathbb{T}$-gain graphs. Unfortunately, in the characterization of equality in Theorem \[Theo.2.6\] for the $\mathbb{T}$-gain graphs, we do not get a nice structure. We introduce a generalization of Hermitian adjacency matrices, for which sharpness of the bounds in Theorem \[Theo.2.6\] can be illustrated nicely. Also conjecture \[conj1\] holds true for these extended classes of matrices. Let $\mathbb{N}$ denote the set of all natural numbers.

Let $ X $ be a digraph and $k\in \mathbb{N} $. The $ k$*-generalized Hermitian adjacency matrix* of $ X $ is denoted by $ H_{k}(X) $ and its $ (s,t)th $ entry is defined by $$h_{st}=\begin{cases}
    1& \text{if both } \mbox{$\overrightarrow{e_{st}}$ \text{and} $\overrightarrow{e_{ts}} \in E(X)$},\\
    e^{\frac{i\pi}{k+1}}& \text{if  } \mbox{$\overrightarrow{e_{st}} \in E(X)$ \text{and} $\overrightarrow{e_{ts}} \notin E(X)$},\\
    e^{-\frac{i\pi}{k+1}}& \text{if  } \mbox{$\overrightarrow{e_{st}} \notin E(X)$ \text{and} $\overrightarrow{e_{ts}} \in E(X)$},\\
    0&\text{otherwise.}\end{cases}$$

For each $ k\in \mathbb{N}$, we can associate a unique matrix $ H_{k}(X) $ with a given digraph $X$. If $ k=1 $, then $ H_1(X) $ is the Hermitian adjacency matrix of $X$. Next, we present a couple of results about $ k $-generalized Hermitian adjacency matrices.

Let $ X $ be a weakly connected digraph. Then for $k\in \mathbb{N}$, $ \lambda_{1}(H_{k}(X)) \leq \rho(H_{k}(X))\leq 3\lambda_{1}(H_{k}(X)) $, where $ \lambda_{1}(H_{k}(X)) $ is the largest eigenvalue of $ H_{k}(X) $.

Let $ G=\Gamma(X) $ be the underlying graph of $ X $. Define a gain function $ \varphi $ on $ G $ by $ \varphi(\overrightarrow{e_{st}})=h_{st} $, where $ h_{st} $ is the $ (s,t)th $ entry of $ H_{k}(X) $. Then $ A(\Phi)=H_{k}(X)$ with $ Im(\varphi) \subseteq \{1,e^{\frac{i\pi}{k+1}},e^{-\frac{i\pi}{k+1}} \} $. Since for any $ k \in \mathbb{N} $, it is clear that $\{1,e^{\frac{i\pi}{k+1}},e^{-\frac{i\pi}{k+1}} \}  \subseteq \{ a+ib \in \mathbb{T}:a \geq 0 \} $. Therefore, by Theorem \[Theo.5.1\], we have $ \lambda_{1}(A(\Phi)) \leq \rho(A(\Phi))\leq 3\lambda_{1}(A(\Phi)) $. That is, $ \lambda_{1}(H_{k}(X)) \leq \rho(H_{k}(X))\leq 3\lambda_{1}(H_{k}(X)) $.

Next theorem is an extension of the characterization given in Theorem \[Theo.2.6\] for the $k$-generalized Hermitian adjacency matrices.

\[Theo5.5\] Let $ X $ be a digraph and $ \Delta $ be the largest vertex degree of $ \Gamma(X) $. Then $ \rho(H_{k}(X)) \leq \Delta $, for all $ k\in \mathbb{N} $. Furthermore, if $ X $ is weakly connected, then equality occur if and only if $ X $ is $ \Delta $-regular and there is a partition of vertex set of $ X $ into $ (2k+2) $ parts ( possibly empty) $ V_{0}, V_{\theta}, \dots, V_{(2k+1)\theta}$, where $ \theta=\frac{\pi}{k+1} $, such that $ X $ has one of the following structures( see Figure \[fig14\]):

(i) **Structure A:** Digraph induced by each vertex set $V_{m\theta} $, for each $ m\in \{0,1, \dots, (2k+1)\}$ contains only digons. For each directed edge $ \overrightarrow{e_{st}} $ of $ X $, if $ v_{s}\in V_{p\theta } $, for some $ p \in \{0,1, \dots, (2k+1)\}$, then $ v_{t}\in V_{((p+1)\theta)~( \mod~2\pi)} $.

(ii) **Structure B:** For every digon $ e_{st} $ in $ X $, if $ v_{s} \in V_{m\theta}$ for some $ m $, then $ v_{t}\in V_{(\pi+m\theta)~( \mod~2\pi)}$. That is, each vertex set $ V_{m\theta } $, for $  m\in \{0,1, \dots, (2k+1)\}$ is an independent set. Every directed edge $ \overrightarrow{e_{st}} $ in $ X $, if $ v_{s}\in V_{m\theta} $ for some $ m $, then $ v_{t}\in V_{(\pi+(1+m)\theta)~( \mod~2\pi)} $.

Let $ X $ be a digraph, and $ G=\Gamma(X) $ be its underlying graph. Fix $ k \in \mathbb{N}$. Define a $ \mathbb{T}$-gain graph $ \Phi $ on $ G $ such that $ \varphi(\overrightarrow{e_{st}})=h_{st} $, where $ h_{st} $ is the $ (s,t)th $ element of $ H_{k}(X) $. Then $ A(\Phi)=H_{k}(X) $. Therefore, by Theorem \[Theo.12\], $\rho(H_{k}(X)) \leq \Delta $. If $ X $ is weakly connected, then $ G $ is connected. Thus, by Theorem \[Theo.4.5\], equality holds if and only if $ G $ is $ \Delta$-regular and either $ \Phi $ is balanced or $ -\Phi $ is balanced. Let us show that $ \Phi $ is balanced if and only if $ X $ has structure A, and $ -\Phi $ is balanced if and only if $ X $ has structure B. Let $ T $ be a normal spanning tree of $ G $ with the root vertex $ v_{1} $, and let $ \theta=\frac{\pi}{k+1} $.

[**Case(i):**]{} Let us show that, if $\Phi$ is balanced, then $X$ has structure A. Without loss of generality, let us assume that $v_{1} \in V_{0}$. Then, for the vertex $v_l$ adjacent to $v_1$ in $T$, three cases arises:

1.  $ v_{l}  \in  V_{\theta} $, if $ \varphi(\overrightarrow{e_{1l}}) = e^{i\theta} $,

2.  $ v_l \in V_{-\theta~(\mod ~ 2\pi)} ,$ if $ \varphi(\overrightarrow{e_{1l}}) =  e^{-i\theta} $, and

3.  $ v_l \in V_{0} $ , if $ \varphi(\overrightarrow{e_{1l}}) =  1. $

Recursively, distribute the vertices of $V(X)$ into the sets $V_{0}, V_{\theta}, V_{2\theta}, \dots ,V_{(2k+1)\theta}$. If $ e_{st}\in E(G)\setminus E(T) $, then there is a fundamental cycle, say $ C $ with respect to $ T $ such that $ e_{st}\in E(C) $. Since $ \Phi $ is balanced, for the directed fundamental cycle $\overrightarrow{C}$ in $\overrightarrow{G}_T$, we have $ \varphi(\overrightarrow{C})=1 $, and hence $ \varphi(\overrightarrow{v_{s}Tv_{t}})$ equals to $\varphi(\overrightarrow{e_{st}})^{-1}$. From the construction of $V_{p\theta} $, for $ p\in \{0,1, \dots, (2k+1)\}$, it is clear that each vertex set $V_{p\theta}$ contains only digons, and for each directed edge $ \overrightarrow{e_{st}} $ of $ X $, if $ v_{s}\in V_{p\theta } $, for some $ p \in \{0,1, \dots, (2k+1)\}$, then $ v_{t}\in V_{(p+1)\theta~( \mod~2\pi)} $.

[**Case(ii)**]{}: Let $ -\Phi $ be balanced. Let $v_1 \in V_0.$ Then, for the vertex $v_l$ adjacent to $v_1$ in $T$, three cases arises:

1.  $ v_{l}  \in  V_{(\pi + \theta) (\mod ~ 2\pi)} $, if $ -\varphi(\overrightarrow{e_{1l}}) = -e^{i \theta} = e^{i(\pi + \theta)} $,

2.  $ v_l \in V_{(\pi -\theta)(\mod ~ 2\pi)} ,$ if $ -\varphi(\overrightarrow{e_{1l}}) = - e^{-i\theta} = e^{i(\pi -\theta)} $, and

3.  $ v_l \in V_{(\pi +0)} $ , if $ -\varphi(\overrightarrow{e_{1l}}) =  -1 =  e^{i\pi} . $

Recursively, distribute the vertices of $V(X)$ into the sets $V_{0}, V_{ \theta}, V_{2\theta}, \dots ,V_{(2k+1)\theta}$. Since $-\Phi$ is balanced, if $ e_{st}\in E(G)\setminus E(T) $, then, as in case(i), $- \varphi(\overrightarrow{v_{s}Tv_{t}})$ equals to $-\varphi(\overrightarrow{e_{st}})^{-1}.$ From the construction, it is clear that, for every digon $ e_{st} $ in $ X $, if $ v_{s} \in V_{m\theta}$ for some $ m $, then $ v_{t}\in V_{(\pi+m\theta)~( \mod~2\pi)}$. Also it follows that for every directed edge $ \overrightarrow{e_{st}} $ in $ X $, if $ v_{s}\in V_{m\theta} $ for some $ m $, then $ v_{t}\in V_{(\pi+(1+m)\theta)~( \mod~2\pi)} $.

Conversely, let $X$ have structure A. Now, let us prove that, if $v_{x} \in V_{g\theta} $ and $ v_{y} \in V_{h\theta}$, then $\varphi(\overrightarrow{v_{x}Tv_{y}})=e^{i(h-g)\theta}.$ It is enough to prove that, if $ v_{x}\in V_{p\theta} $ and $ v_{y}\in V_{(p+1)\theta} $, for some $ p \in \{ 0, 1, \dots, (2k+1)\} $, then $\varphi(\overrightarrow{v_{x}Tv_{y}})=e^{i\theta}$. Let $ \overrightarrow{v_{x}Tv_{y}}= v_{x}\overrightarrow{e_{xs}}v_{s}\cdots v_t\overrightarrow{e_{ty}}v_{y}$. From case(i), we have the following:

(a) $ v_{s}\in V_{(p + 1)\theta~ (\mod ~ 2\pi)} $ if and only if $ \varphi(\overrightarrow{e_{xs}}) = e^{i\theta}$,

(b) $ v_{s}\in V_{(p -1)\theta~ (\mod ~ 2\pi)} $ if and only if $\varphi(\overrightarrow{e_{xs}}) = e^{-i\theta}$, and

(c) $ v_{s}\in V_{p\theta} $ if and only if $ \varphi(\overrightarrow{e_{xs}}) = e^{i0}=1$.

If $\varphi(\overrightarrow{e_{xs}})=e^{i\theta}$, then $ v_{s}\in V_{(p+ 1)\theta} $. Suppose the rest of the vertices in the path lie inside $V_{(p+1)\theta}$, then gain of every other edge in the path is $1$, and hence $\varphi(\overrightarrow{v_{x}Tv_{y}})=e^{i((p+1)-p)\theta} = e^{i\theta}.$ Suppose some of the vertices of the path $\overrightarrow{v_{x}Tv_{y}}$ lie outside $V_{(p+1)\theta}$, then whenever we have an outgoing edge from $V_{(p+1)\theta}$ to $V_{p\theta}$ or to $V_{(p+2)\theta}$, we have an incoming edge from the same. Thus the product of the gains of these two oriented edges is $1$. Hence, in this case also we have $\varphi(\overrightarrow{v_{x}Tv_{y}}) =  e^{i\theta}.$ Inductively, we can complete the proof of this case. The proof of the other cases is similar.

Let $ e_{xy}\in E(G)\setminus E(T) $. Then there is a directed fundamental cycle $ \overrightarrow{C} $ in $ \overrightarrow{G}_{T} $ such that $ \overrightarrow{e_{xy}}\in E(\overrightarrow{C}) $. Suppose $ v_{y}\in V_{p \theta} $. Since the possible values of $ \varphi(\overrightarrow{e_{xy}})$ are $ e^{i\theta}$, $e^{-i \theta}$ or $ e^{i0} $. So $ v_{x} \in V_{(p-1)\theta ~(\mod ~ 2\pi)} $, $v_{x} \in  V_{(p+1)\theta ~(\mod ~ 2\pi)} $ or $ v_{x}\in V_{p\theta} $. If $ \varphi(\overrightarrow{e_{xy}})=e^{i\theta}$, then $ v_{x} \in V_{(p-1)\theta ~(\mod ~ 2\pi)}   $. Now, $ \varphi(\overrightarrow{v_{y}Tv_{x}})=e^{i((p-1)-p)\theta} = e^{-i\theta}$. Therefore, $ \varphi(\overrightarrow{C})= \varphi(\overrightarrow{v_{y}Tv_{x}})\varphi(\overrightarrow{e_{xy}})=e^{-i\theta +i \theta}=1$. Thus $ \varphi(\overrightarrow{C})=1 $, and all the directed fundamental cycles in $ \overrightarrow{G}_{T} $ are neutral. Therefore, $ \Phi $ is balanced.

Similarly, we can prove that, if $X$ has structure B, then $-\Phi$ is balanced.

Next, we illustrate the structure A and structure B corresponding to $k$-generalized Hermitian adjacency matrices pictorially for various positive integers $k$.

![Structure A and Structure B for $ H(X), H_{2}(X), H_{3}(X) $ and $ H_{k}(X) $ []{data-label="fig14"}](fig12.pdf)

Acknowledgments {#acknowledgments .unnumbered}
===============

Aniruddha Samanta thanks University Grants Commission(UGC) for the financial support in the form of the Senior Research Fellowship (Ref.No: 19/06/2016(i)EU-V).M. Rajesh Kannan thanks the Department of Science and Technology, India, for financial support through the Early Career Research Award (ECR/2017/000643).

[^1]: aniruddha.sam@gmail.com

[^2]: rajeshkannan1.m@gmail.com, rajeshkannan@maths.iitkgp.ac.in
