Achievable multiplicity partitions in the inverse eigenvalue problem of a graph

TL;DR

This paper investigates the inverse eigenvalue problem for graphs, specifically focusing on which graphs allow matrices with eigenvalue multiplicities partitioned into two parts, and provides methods to construct such graphs.

Contribution

It extends the characterization of graphs with two-part eigenvalue multiplicity partitions to broader classes, including generalizations of complete multipartite graphs.

Findings

Identified families of graphs with multiplicity partition [n-k,k]

Developed methods to construct graphs with specified eigenvalue multiplicity partitions

Demonstrated the complexity of characterizing such graphs

Abstract

Associated to a graph $G$ is a set $\mathcal{S}(G)$ of all real-valued symmetric matrices whose off-diagonal entries are nonzero precisely when the corresponding vertices of the graph are adjacent, and the diagonal entries are free to be chosen. If $G$ has $n$ vertices, then the multiplicities of the eigenvalues of any matrix in $\mathcal{S}(G)$ partition $n$; this is called a multiplicity partition. We study graphs for which a multiplicity partition with only two integers is possible. The graphs $G$ for which there is a matrix in $\mathcal{S}(G)$ with partitions $[n-2,2]$ have been characterized. We find families of graphs $G$ for which there is a matrix in $\mathcal{S}(G)$ with multiplicity partition $[n-k,k]$ for $k\geq 2$. We focus on generalizations of the complete multipartite graphs. We provide some methods to construct families of graphs with given multiplicity partitions starting from smaller such graphs. We also give constructions for graphs with matrix in $\mathcal{S}(G)$ with multiplicity partition $[n-k,k]$ to show the complexities of characterizing these graphs.