Two Distinct Eigenvalues from a New Graph Product

TL;DR

This paper introduces a new graph product method that constructs infinite families of graphs with exactly two distinct eigenvalues, expanding understanding of spectral graph properties.

Contribution

The paper presents a novel graph product that generates graphs with two eigenvalues, unifying and extending known classes of such graphs.

Findings

Constructs infinite families of graphs with q(G)=2

Shows existing q(G)=2 graphs can be derived from the new product

Provides a framework for analyzing spectral properties of graphs

Abstract

The parameter $q(G)$ of a graph $G$ is the minimum number of distinct eigenvalues of a symmetric matrix whose pattern is given by $G$. We introduce a novel graph product by which we construct new infinite families of graphs that achieve $q(G)=2$. Several graph families for which it is already known that $q(G)=2$ can also be thought of as arising from this new product.