Minimum number of distinct eigenvalues of graphs

TL;DR

This paper investigates the minimum number of distinct eigenvalues of graphs, establishing bounds, exploring specific graph families, and analyzing how graph modifications affect this parameter.

Contribution

It introduces bounds for the parameter $q(G)$, characterizes graphs with $q(G)=2$, and examines how edge and vertex changes impact $q(G)$.

Findings

Many graphs have $q(G)=2$

Adding or removing edges/vertices can significantly alter $q(G)$

Graphs with $q(G)$ near the number of vertices are related to small maximum multiplicity families

Abstract

The minimum number of distinct eigenvalues, taken over all real symmetric matrices compatible with a given graph $G$, is denoted by $q(G)$. Using other parameters related to $G$, bounds for $q(G)$ are proven and then applied to deduce further properties of $q(G)$. It is shown that there is a great number of graphs $G$ for which $q(G)=2$. For some families of graphs, such as the join of a graph with itself, complete bipartite graphs, and cycles, this minimum value is obtained. Moreover, examples of graphs $G$ are provided to show that adding and deleting edges or vertices can dramatically change the value of $q(G)$. Finally, the set of graphs $G$ with $q(G)$ near the number of vertices is shown to be a subset of known families of graphs with small maximum multiplicity.