Relation between the inertia indices of a complex unit gain graph and those of its underlying graph. Issue 5 (24th March 2022)

• Record Type: Journal Article
• Authors: Zaman, Shahid; He, Xiaocong
• Is Part Of: Linear & multilinear algebra. Volume 70:Issue 5(2022)
• Page Start: 843
• Page End: 877
• Publication Date: 2022-03-24
• Subjects: T-gain graph -- positive inertia index -- negative inertia index -- dimension of cycle space; 05C75; Algebras, Linear -- Periodicals; Multilinear algebra -- Periodicals; 512.505
• DOI: 10.1080/03081087.2020.1749224
• Languages: English
• ISSNs: 0308-1087

Abstract

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.