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

• DOI: 10.48550/arxiv.1412.5727
• Publication Date: 2014-01-01
• Authors: Xiaolin Chen; Xueliang Li; Huishu Lian
• Keywords: Spectral Radius

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