On cospectrality of gain graphs

Abstract

Abstract We define <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G -cospectrality of two <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G -gain graphs <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo>,</m:mo> <m:mi>ψ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(\Gamma ,\psi ) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="normal">Γ</m:mi> <m:mo accent="false">′</m:mo> <m:mo>,</m:mo> <m:mi>ψ</m:mi> <m:mo accent="false">′</m:mo> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(\Gamma ^{\prime} ,\psi ^{\prime} ) , proving that it is a switching isomorphism invariant. When <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G is a finite group, we prove that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G -cospectrality is equivalent to cospectrality with respect to all unitary representations of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>v</m:mi> </m:math> v can be simultaneously conjugated. In particular, the number of switching equivalence classes on an underlying graph <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="normal">Γ</m:mi> </m:math> \Gamma with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>n</m:mi> </m:math> n vertices and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>m</m:mi> </m:math> m edges, is equal to the number of simultaneous conjugacy classes of the group <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>G</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> <m:mo>−</m:mo> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> </m:math> {G}^{m-n+1} . We provide examples of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G -cospectral switching nonisomorphic graphs and we prove that any gain graph on a cycle is determined by its <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> G -spectrum. Moreover, we show that when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> 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.

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