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arXiv · 2306.14127

Laplacain eigenvalue distribution and diameter of graphs

Leyou Xu·Bo Zhou
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Abstract

Let $G$ be a connected graph on $n$ vertices with diameter $d$. It is known that if $2\le d\le n-2$, there are at most $n-d$ Laplacian eigenvalues in the interval $[n-d+2, n]$. In this paper, we show that if $1\le d\le n-3$, there are at most $n-d+1$ Laplacian eigenvalues in the interval $[n-d+1, n]$. Moreover, we try to identify the connected graphs on $n$ vertices with diameter $d$, where $2\le d\le n-3$, such that there are at most $n-d$ Laplacian eigenvalues in the interval $[n-d+1, n]$.

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Leyou Xu, Bo Zhou. 2023-06-25. Laplacain eigenvalue distribution and diameter of graphs. https://arxiv.org/abs/2306.14127

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