Quantum graphs are metric graphs equipped with a differential (or pseudo-differential) operator acting on functions on the graph. A function f on a metric graph is defined as the | E | -tuple of functions f e ( x e ) on the intervals. The Hilbert space of the graph is ⨁ ⨁ e ∈ ∈ E L 2 ( [ 0 , L e ] ) where the inner product of two functions is