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

Abstract

Let $G$ be a simple connected graph with vertex set $V(G)=\{v_1,v_2,\dots ,v_n\}$. 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 $-\frac{1}{2}$.