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arXiv · 2506.00921

Girth and Laplacian eigenvalue distribution

Leyou Xu·Bo Zhou
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Abstract

Let $G$ be a connected graph of order $n$ with girth $g$. For $k=1,\dots,\min\{g-1, n-g\}$, let $n(G,k)$ be the number of Laplacian eigenvalues (counting multiplicities) of $G$ that fall inside the interval $[n-g-k+4,n]$. We prove that if $g\ge 4$, then \[ n(G,k)\le n-g. \] Those graphs achieving the bound for $k=1,2$ are determined. We also determine the graphs $G$ with $g=3$ such that $n(G,k)=n-1, n-2, n-3$.

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Leyou Xu, Bo Zhou. 2025-06-01. Girth and Laplacian eigenvalue distribution. https://arxiv.org/abs/2506.00921

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