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Mathematics > Combinatorics

arXiv:2204.09884 (math)
[Submitted on 21 Apr 2022 (v1), last revised 26 Apr 2022 (this version, v3)]

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

Authors:Yongtao Li, Yuejian Peng
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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.
Comments: 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
Subjects: Combinatorics (math.CO); Spectral Theory (math.SP)
MSC classes: 05C50, 05C35
Cite as: arXiv:2204.09884 [math.CO]
  (or arXiv:2204.09884v3 [math.CO] for this version)
  https://doi.org/10.48550/arXiv.2204.09884
Journal reference: The Electronic Journal of Combinatorics 29(4) (2022), #P4.2
Related DOI: https://doi.org/10.37236/11236

Submission history

From: Yongtao Li [view email]
[v1] Thu, 21 Apr 2022 05:16:02 UTC (1,500 KB)
[v2] Sat, 23 Apr 2022 09:03:40 UTC (1 KB) (withdrawn)
[v3] Tue, 26 Apr 2022 02:22:13 UTC (1,500 KB)
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