On the second largest distance eigenvalue of a graph

Abstract

Let $G$ be a simple connected graph of order $n$ and $D(G)$ be the distance matrix of $G.$ Suppose that ${\lambda }_1(D(G))\ge {\lambda }_2(D(G))\ge \cdots \ge {\lambda }_n(D(G))$ are the distance spectrum of $G$. A graph $G$ is said to be determined by its $D$-spectrum if with respect to the distance matrix $D(G)$, any graph with the same spectrum as $G$ is isomorphic to $G$. In this paper, we consider spectral characterization on the second largest distance eigenvalue ${\lambda }_2(D(G))$ of graphs, and prove that the graphs with ${\lambda }_2(D(G))\le \frac{17-\sqrt{329}}{2}\approx -0.5692$ are determined by their $D$-spectra.