On the second largest distance eigenvalue of a graph

TL;DR

This paper investigates the spectral properties of graphs related to their distance matrices, specifically focusing on the second largest eigenvalue, and proves that graphs with a certain bound on this eigenvalue are uniquely identified by their distance spectrum.

Contribution

It establishes a spectral characterization for graphs based on the second largest distance eigenvalue, identifying a specific bound that guarantees spectral uniqueness.

Findings

Graphs with $ ext{lambda}_2(D(G)) ext{ below } -0.5692$ are uniquely determined by their $D$-spectra.

The paper provides a spectral criterion for graph identification based on the second largest distance eigenvalue.

Abstract

Let $G$ be a simple connected graph of order $n$ and $D(G)$ be the distance matrix of $G.$ Suppose that $\lambda_{1}(D(G))\geq\lambda_{2}(D(G))\geq\cdots\geq\lambda_{n}(D(G))$ 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 $\lambda_{2}(D(G))\leq\frac{17-\sqrt{329}}{2}\approx-0.5692$ are determined by their $D$-spectra.