Graphs with at most three distance eigenvalues different from $-1$ and $-2$

Abstract

Let $G$ be a connected graph on $n$ vertices, and let $D(G)$ be the distance matrix of $G$. Let ${\partial }_1(G)\ge {\partial }_2(G)\ge \cdots \ge {\partial }_n(G)$ denote the eigenvalues of $D(G)$. In this paper, we characterize all connected graphs with ${\partial }_3(G)\le -1$ and ${\partial }_{n-1}(G)\ge -2$. By the way, we determine all connected graphs with at most three distance eigenvalues different from $-1$ and $-2$.

17 pages, 3 figures