Graphs with small diameter determined by their $D$-spectra

Abstract

Let $G$ be a connected graph with vertex set $V(G)=\{v_1,v_2,...,v_n\}$. The distance matrix $D(G)=(d_{ij}{)}_{n\times n}$ is the matrix indexed by the vertices of $G,$ where $d_{ij}$ denotes the distance between the vertices $v_i$ and $v_j$. Suppose that ${\lambda }_1(D)\ge {\lambda }_2(D)\ge \cdots \ge {\lambda }_n(D)$ are the distance spectrum of $G$. The graph $G$ is said to be determined by its $D$-spectrum if with respect to the distance matrix $D(G)$, any graph having the same spectrum as $G$ is isomorphic to $G$. In this paper, we give the distance characteristic polynomial of some graphs with small diameter, and also prove that these graphs are determined by their $D$-spectra.