Graphs with three distinct distance eigenvalues

TL;DR

This paper investigates the spectral properties of graphs with three distinct distance eigenvalues, constructing specific tree families and characterizing certain graphs to address open problems in spectral graph theory.

Contribution

It introduces a recursive construction of trees with a specific eigenvalue and characterizes all trees with three distinct distance eigenvalues, partially solving an open problem.

Findings

Constructed an infinite family of trees with distance eigenvalue -1

Characterized all ree connected graphs with three distinct distance eigenvalues and smallest eigenvalue -3

Provided a complete characterization of trees with three distinct distance eigenvalues

Abstract

In this paper, some special distance spectral properties of graphs are considered. Concretely, we recursively construct an infinite family of trees with distance eigenvalue $-1$, and determine all $\{C_3,C_4\}$-free connected graphs with three distinct distance eigenvalues of which the smallest one is equal to $-3$, which partially answers a problem posed by Koolen, Hayat and Iqbal [Linear Algebra Appl. 505 (2016) 97--108]. Furthermore, we characterize all trees with three distinct distance eigenvalues.