Tricyclic graphs for which the second largest distance eigenvalue less than $-\frac{1}{2}$

Kexin Yang; Ligong Wang

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arXiv · 2509.12640

Tricyclic graphs for which the second largest distance eigenvalue less than $-\frac{1}{2}$

Kexin Yang·Ligong Wang

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Abstract

Let $G$ be a simple connected graph with vertex set $V(G)=\{v_{1}, v_{2}, \ldots, v_{n}\}$. The distance $d_G(v_i,v_j)$ between two vertices $v_i$ and $v_j$ of $G$ is the length of a shortest path between $v_i$ and $v_j$. The distance matrix of $G$ is defined as $D(G)=(d_G(v_i,v_j))_{n\times n}$. The second largest distance eigenvalue of $ G $ is the second largest eigenvalues of $D(G)$. Guo and Zhou [Discrete Math. 347(2024), 114082] proved that any connected graph with the second largest distance eigenvalue less than $-\frac{1}{2}$ is chordal, and characterize all bicyclic graphs and split graphs with the second largest distance eigenvalue less than $-\frac{1}{2}$. Based on this, we characterize all tricyclic graphs with the second largest distance eigenvalue less than $-\frac{1}{2}$.

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Kexin Yang, Ligong Wang. 2025-09-16. Tricyclic graphs for which the second largest distance eigenvalue less than $-\frac{1}{2}$. https://arxiv.org/abs/2509.12640

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