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Mathematics > Combinatorics
Title: The maximum spectral radius of non-bipartite graphs forbidding short odd cycles
(Submitted on 21 Apr 2022 (v1), last revised 26 Apr 2022 (this version, v3))
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.
Submission history
From: Yongtao Li [view email][v1] Thu, 21 Apr 2022 05:16:02 GMT (1500kb,D)
[v2] Sat, 23 Apr 2022 09:03:40 GMT (0kb,I)
[v3] Tue, 26 Apr 2022 02:22:13 GMT (1500kb,D)
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