Spectral radius of graphs with given size and odd girth

TL;DR

This paper determines the maximum spectral radius of graphs with a fixed size and odd girth when the number of edges is odd, revealing structural properties and settling a previous conjecture.

Contribution

It identifies the extremal graphs maximizing spectral radius under size and odd girth constraints, and introduces a spectral threshold related to odd cycles.

Findings

The extremal graph is characterized as a subdivided complete bipartite graph.

A spectral radius threshold ta(m) is established for the existence of short odd cycles.

The results settle a conjecture and extend previous findings in spectral graph theory.

Abstract

Let $\mathcal{G}(m,k)$ be the set of graphs with size $m$ and odd girth (the length of shortest odd cycle) $k$. In this paper, we determine the graph maximizing the spectral radius among $\mathcal{G}(m,k)$ when $m$ is odd. As byproducts, we show that, there is a number $\eta(m)>\sqrt{m-k+3}$ such that every non-bipartite graph $G$ with size $m$ and spectral radius $\rho\ge \eta(m)$ must contains an odd cycle of length less than $k$ unless $m$ is odd and $G\cong SK_{k,m}$, which is the graph obtained by subdividing an edge $k-2$ times of complete bipartite $K_{2,\frac{m-k+2}{2}}$. This result implies the main results of [Discrete Math. 345 (2022)] and \cite{li-peng}, and settles the conjecture in \cite{li-peng} as well.