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

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.

27 pages, 6 figures. We would like to express sincere thanks to Huiqiu Lin, Bo Ning and Mingqing Zhai for kind discussions, which considerably improves the presentation of the manuscript. The present work can be viewed as the second paper of our previous project arXiv:2204.09194. Any comments and suggestions are welcome

References in corpus (2)

• The maximum spectral radius of C_4-free graphs of given order and size
• New proofs of stability theorems on spectral graph problems

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• Refinement on spectral Turán's theorem
• A spectral extremal problem on non-bipartite triangle-free graphs
• Spectral extrema of graphs with fixed size: forbidden a fan graph, friendship graph or theta graph
• Spectral extremal problems for non-bipartite graphs without odd cycles