The graphs with exactly two distance eigenvalues different from $-1$ and $-3$

TL;DR

This paper characterizes graphs with specific distance eigenvalue constraints, identifying three classes of such graphs and proving the friendship graph is uniquely determined by its distance spectrum.

Contribution

It provides a complete characterization of graphs with exactly two distance eigenvalues different from -1 and -3, and shows the friendship graph is determined by its distance spectrum.

Findings

Identified three infinite classes of graphs with the specified eigenvalue properties.

Proved that these graphs are uniquely determined by their distance spectra.

Established that the friendship graph is uniquely identified by its distance spectrum.

Abstract

In this paper, we completely characterize the graphs with third largest distance eigenvalue at most $-1$ and smallest distance eigenvalue at least $-3$. In particular, we determine all graphs whose distance matrices have exactly two eigenvalues (counting multiplicity) different from $-1$ and $-3$. It turns out that such graphs consist of three infinite classes, and all of them are determined by their distance spectra. We also show that the friendship graph is determined by its distance spectrum.