underlying graph of a quiver

Let Q=(Q0,Q1,s,t) be a quiver, i.e. Q0 is a set of vertices, Q1 is a set of arrows and s,t:Q1Q0 are functions which take each arrow to its source and target respectively.

Definition. An underlying graph of Q or graph associated with Q is a graph


such that V=Q0, E=Q1 and τ:EVsym2 is given by


In other words G is a graph which is obtained from Q after forgeting the orientation of arrows. The definition of a graph used here is taken from this entry (http://planetmath.org/AlternativeDefinitionOfAMultigraph).

Note, that if we know the underlying graph G of a quiver Q, then the information we have is not enough to reconstruct Q (except for a trivial case with no edges). The orientation of arrows is lost forever. In some cases it is possible to reconstruct Q up to an isomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath of quivers (http://planetmath.org/MorphismsBetweenQuivers), for example graph


uniquely (up to isomorphism) determines its quiver, but


does not uniquely determine its quiver. Indeed, there are exactly two nonisomorphic quivers with underlying graph G, namely:

Title underlying graph of a quiver
Canonical name UnderlyingGraphOfAQuiver
Date of creation 2013-03-22 19:16:59
Last modified on 2013-03-22 19:16:59
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Definition
Classification msc 14L24