On regular graphs with four distinct eigenvalues

TL;DR

This paper investigates the structure of regular graphs with four eigenvalues, proving non-existence results for certain configurations and characterizing specific classes of such graphs, showing they are uniquely determined by their spectra.

Contribution

It establishes the non-existence of connected regular graphs with four eigenvalues having three simple eigenvalues and fully characterizes certain subclasses of these graphs.

Findings

Non-existence of graphs with three simple eigenvalues among four eigenvalues

Complete characterization of graphs in (4,2,-1)

Graphs in (4,-1) and (4,2,0) are determined by their spectra

Abstract

Let $\mathcal{G}(4,2)$ be the set of connected regular graphs with four distinct eigenvalues in which exactly two eigenvalues are simple, $\mathcal{G}(4,2,-1)$ (resp. $\mathcal{G}(4,2,0)$) the set of graphs belonging to $\mathcal{G}(4,2)$ with $-1$ (resp. $0$) as an eigenvalue, and $\mathcal{G}(4,\geq -1)$ the set of connected regular graphs with four distinct eigenvalues and second least eigenvalue not less than $-1$. In this paper, we prove the non-existence of connected graphs having four distinct eigenvalues in which at least three eigenvalues are simple, and determine all the graphs in $\mathcal{G}(4,2,-1)$. As a by-product of this work, we characterize all the graphs belonging to $\mathcal{G}(4,\geq-1)$ and $\mathcal{G}(4,2,0)$, respectively, and show that all these graphs are determined by their spectra.