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arXiv · 2504.15772

Laplacian eigenvalue distribution and girth of graphs

Wenhao Zhen·Dein Wong·Songnian Xu
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Abstract

Let $G$ be a connected graph on $n$ vertices with girth $g$. Let $m_GI$ denote the number of Laplacian eigenvalues of graph $G$ in an interval $I$. In this paper, we show that if $G$ is not a cycle, then $m_G(n-g+3,n]\leq n-g$. Moreover, we prove that $m_G(n-g+3,n]= n-g$ if and only if $G\cong C_3$ or $G\cong K_{3,2}$ or $G\cong U_1$, where $U_1$ is obtained from a cycle by joining a single vertex with a vertex of this cycle.

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Wenhao Zhen, Dein Wong, Songnian Xu. 2025-06-22. Laplacian eigenvalue distribution and girth of graphs. https://arxiv.org/abs/2504.15772

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