Inertia indices of a complex unit gain graph in terms of matching number

2021/08/01 by Yong Lu, Qi Wu, Lu, Yong +1
Computer Science · Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #Random Matrices and Applications

paper · pdf · doi:10.48550/arxiv.2108.01443

openalex publication_date 2021/08/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A complex unit gain graph is a triple φ=(G, \mathbbT, φ) (or G^φ for short) consisting of a simple graph G, as the underlying graph of G^φ, the set of unit complex numbers \mathbbT=z∈ ℂ: |z| = 1 and a gain function φ: \overrightarrowE→ \mathbbT such that φ(e_i,j)=φ(e_j,i) ^-1. Let A(G^φ) be adjacency matrix of G^φ. In this paper, we prove that m(G)-c(G)≤ p(G^φ)≤ m(G)+c(G), m(G)-c(G)≤ n(G^φ)≤ m(G)+c(G), where p(G^φ), n(G^φ), m(G) and c(G) are the number of positive eigenvalues of A(G^φ), the number of negative eigenvalues of A(G^φ), the matching number and the cyclomatic number of G, respectively. Furthermore, we characterize the graphs which attain the upper bounds and the lower bounds, respectively.

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