Solution to a conjecture on the maximum skew-spectral radius of odd-cycle graphs

2014/12/18 by Xiaolin Chen, Xueliang Li, Chen, Xiaolin +3
Computer Science · Mathematics · #05C20 #05C50 #05C90 #Advanced Graph Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Matrix Theory and Algorithms #math.CO #msc:05C20 #msc:05C50 #msc:05C90

paper · pdf · doi:10.48550/arxiv.1412.5727

14 pages

arxiv created 2014/12/18 · openalex publication_date 2014/12/18 · arxiv updated 2014/12/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let G be a simple graph with no even cycle, called an odd-cycle graph. Cavers et al. [Cavers et al. Skew-adjacency matrices of graphs, Linear Algebra Appl. 436(2012), 4512--1829] showed that the spectral radius of G^σ is the same for every orientation σ of G, and equals the maximum matching root of G. They proposed a conjecture that the graphs which attain the maximum skew spectral radius among the odd-cycle graphs G of order n are isomorphic to the odd-cycle graph with one vertex degree n-1 and size m=\lfloor 3(n-1)/2\rfloor. This paper, by using the Kelmans transformation, gives a proof of the conjecture. Moreover, sharp upper bounds of the maximum matching roots of the odd-cycle graphs with given order n and size m are given and extremal graphs are characterized.

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