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

A T-gain graph is a triple Phi = (G, T, phi) consisting of an underlying graph G = (V(G), E(G)), the circle group T = {z is an element of C : vertical bar z vertical bar = 1} and a gain function phi : (E) over right arrow -> T, such that phi(e(ij)) = phi(e(ji))(-1) = (phi(e(ji)) over bar. In this paper, we focus our attention on the relations between the inertia indices of T-gain graph Phi and the inertia indices of its underlying graph G. We obtain sharp lower and upper bounds on p(Phi) (resp., n(Phi)) 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.

Automated summary

This paper focuses on the relations between the inertia indices of a T-gain graph and the inertia indices of its underlying graph. It provides sharp lower and upper bounds on the inertia indices and characterizes the corresponding extremal T-gain graphs and signed graphs.