On cospectrality of gain graphs

Abstract

Abstract We define G G -cospectrality of two G G -gain graphs ( Γ , ψ ) \left(\Gamma ,\psi ) and ( Γ ′ , ψ ′ ) \left(\Gamma ^{\prime} ,\psi ^{\prime} ) , proving that it is a switching isomorphism invariant. When G G is a finite group, we prove that G G -cospectrality is equivalent to cospectrality with respect to all unitary representations of G 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 v can be simultaneously conjugated. In particular, the number of switching equivalence classes on an underlying graph Γ \Gamma with n n vertices and m m edges, is equal to the number of simultaneous conjugacy classes of the group G m − n + 1 {G}^{m-n+1} . We provide examples of G G -cospectral switching nonisomorphic graphs and we prove that any gain graph on a cycle is determined by its G G -spectrum. Moreover, we show that when G 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.