A note on graphs with exactly two main eigenvalues

TL;DR

This paper explores connected graphs with exactly two main eigenvalues, providing new constructions and demonstrating the existence of such graphs with an unbounded number of distinct valencies.

Contribution

It introduces several constructions for these graphs and shows that they can have arbitrarily many distinct valencies, expanding understanding of their structural diversity.

Findings

Constructed new families of graphs with two main eigenvalues.

Proved the existence of graphs with unbounded valencies.

Enhanced classification of graphs based on eigenvalue properties.

Abstract

In this note, we consider connected graphs with exactly two main eigenvalues. We will give several constructions for them, and as a consequence we show a family of those graphs with an unbounded number of distinct valencies.