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

TL;DR

This paper proves a conjecture that identifies the extremal odd-cycle graphs with maximum skew-spectral radius, using Kelmans transformation, and establishes bounds for maximum matching roots based on graph order and size.

Contribution

It provides a proof of the conjecture on maximum skew-spectral radius for odd-cycle graphs and characterizes extremal graphs with sharp bounds.

Findings

Confirmed the conjecture about extremal odd-cycle graphs.

Derived sharp upper bounds for maximum matching roots.

Characterized extremal graphs achieving these bounds.

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 $G^\sigma$ is the same for every orientation $\sigma$ 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.