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

TL;DR

This paper investigates graphs with small diameter, providing their distance characteristic polynomials and proving that certain such graphs are uniquely identified by their $D$-spectra.

Contribution

It derives the distance characteristic polynomial for specific small-diameter graphs and proves these graphs are uniquely determined by their $D$-spectra.

Findings

Distance characteristic polynomials for some small-diameter graphs are obtained.

Certain graphs with small diameter are uniquely 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 $\lambda_{1}(D)\geq\lambda_{2}(D)\geq\cdots\geq\lambda_{n}(D)$ 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.