On the minimum number of distinct eigenvalues of a threshold graph

TL;DR

This paper characterizes connected threshold graphs with minimal eigenvalue diversity, determines specific eigenvalue counts for certain graph classes, and provides bounds for the minimum number of distinct eigenvalues in these graphs.

Contribution

It offers a complete characterization of connected threshold graphs with two eigenvalues and bounds for the minimum number of eigenvalues across all such graphs.

Findings

Characterization of connected threshold graphs with q(G)=2.

Determination of q(G) for graphs with specific trace values.

Sharp upper bound for q(G) in connected threshold graphs.

Abstract

For a graph $G$, we associate a family of real symmetric matrices, $S(G)$, where for any $A\in S(G)$, the location of the nonzero off-diagonal entries of $A$ are governed by the adjacency structure of $G$. Let $q(G)$ be the minimum number of distinct eigenvalues over all matrices in $S(G)$. In this work, we give a characterization of all connected threshold graphs $G$ with $q(G)=2$. Moreover, we study the values of $q(G)$ for connected threshold graphs with trace $2$, $3$, $n-2$, $n-3$, where $n$ is the order of threshold graph. The values of $q(G)$ are determined for all connected threshold graphs with $7$ and $8$ vertices with two exceptions. Finally, a sharp upper bound for $q(G)$ over all connected threshold graph $G$ is given.