Relation between the inertia indices of a complex unit gain graph and those of its underlying graph

Abstract

A T-gain graph is a triple Φ=(G,T,ϕ) consisting of an underlying graph G=(V(G),E(G)), the circle group T={z∈C:|z|=1} and a gain function ϕ:E→→T, such that ϕ(eij)=ϕ(eji)−1=ϕ(eji)¯. In this paper, we focus our attention on the relations between the inertia indices of T-gain graph Φ and the inertia indices of its underlying graph G. We obtain sharp lower and upper bounds on p(Φ) (resp., n(Φ)) in terms of p(G) (resp., n(G)) and characterize those corresponding extremal T-gain graphs, respectively. As a corollary, we also characterize those signed graphs whose inertia indices obtaining the sharp bounds, respectively.

Topics & Concepts