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arXiv · 2307.00917

A graph for which the second largest distance eigenvalue is less than $\frac{-3+\sqrt{5}}{2}$ is chordal

Haiyan Guo·Bo Zhou
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Abstract

Let $G$ be a connected graph with vertex set $V(G)$. The distance, $d_G(u,v)$, between vertices $u$ and $v$ in $G$ is defined as the length of a shortest path between $u$ and $v$ in $G$. The distance matrix of $G$ is the matrix $D(G)=(d_G(u,v))_{u,v\in V(G)}$. The second largest distance eigenvalue of $G$ is the second largest one in the spectrum of $D(G)$. We show that any connected graph with the second largest distance eigenvalue less than $\frac{-3+\sqrt{5}}{2}$ is chordal, and characterize those bicyclic graphs and split graphs with the second largest distance eigenvalue less than $-\frac{1}{2}$.

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Haiyan Guo, Bo Zhou. 2023-07-03. A graph for which the second largest distance eigenvalue is less than $\frac{-3+\sqrt{5}}{2}$ is chordal. https://arxiv.org/abs/2307.00917

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