Biregular graphs with three eigenvalues

TL;DR

This paper studies nonregular graphs with exactly three eigenvalues, focusing on those with two valencies, providing new constructions, structure theorems, and a rare example with three valencies.

Contribution

It introduces new examples and classifications of nonregular graphs with three eigenvalues, especially those with two or three valencies, advancing understanding of their structure.

Findings

Constructed new examples of such graphs

Developed structure theorems and valency constraints

Classified certain families of these graphs

Abstract

We consider nonregular graphs having precisely three distinct eigenvalues. The focus is mainly on the case of graphs having two distinct valencies and our results include constructions of new examples, structure theorems, valency constraints, and a classification of certain special families of such graphs. We also present a new example of a graph with three valencies and three eigenvalues of which there are currently only finitely many known examples.