Relation between the inertia indices of a complex unit gain graph and those of its underlying graph. Issue 5 (24th March 2022) - Zaman, Shahid; He, Xiaocong
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 ( e i j ) = ϕ ( e j i ) − 1 = ϕ ( e j i ) ¯ . 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.