On cospectrality of gain graphs

Abstract

We define G-cospectrality of two G-gain graphs (Γ, ψ) and (Γ ′ , ψ ′ ), proving that it is a switching isomorphism invariant. When G is a finite group, we prove that G-cospectrality is equivalent to cospectrality with respect to all unitary representations of G. Moreover, we show that two connected gain graphs are switching equivalent if and only if the gains of their closed walks centered at an arbitrary vertex v can be simultaneously conjugated. In particular, the number of switching equivalence classes on an underlying graph Γ with n vertices and m edges, is equal to the number of simultaneous conjugacy classes of the group G m−n+1 .We provide examples of G-cospectral non-switching isomorphic graphs and we prove that any gain graph on a cycle is determined by its G-spectrum. Moreover, we show that when G is a finite cyclic group, the cospectrality with respect to a faithful irreducible representation implies the cospectrality with respect to any other faithful irreducible representation, and that the same assertion is false in general.