The maximum spectral radius of non-bipartite graphs forbidding short odd cycles

TL;DR

This paper investigates the maximum spectral radius of non-bipartite graphs that do not contain short odd cycles, providing new proofs and refinements of existing bounds using an alternative methodological approach.

Contribution

It introduces an alternative proof technique for bounds on spectral radius in non-bipartite graphs without short odd cycles, refining previous results.

Findings

Established a new proof method for spectral radius bounds

Refined bounds for non-bipartite graphs without short odd cycles

Enhanced understanding of spectral properties related to odd cycle restrictions

Abstract

It is well-known that eigenvalues of graphs can be used to describe structural properties and parameters of graphs. A theorem of Nosal states that if $G$ is a triangle-free graph with $m$ edges, then $\lambda (G)\le \sqrt{m}$, equality holds if and only if $G$ is a complete bipartite graph. Recently, Lin, Ning and Wu [Combin. Probab. Comput. 30 (2021)] proved a generalization for non-bipartite triangle-free graphs. Moreover, Zhai and Shu [Discrete Math. 345 (2022)] presented a further improvement. In this paper, we present an alternative method for proving the improvement by Zhai and Shu. Furthermore, the method can allow us to give a refinement on the result of Zhai and Shu for non-bipartite graphs without short odd cycles.