Ordered multiplicity inverse eigenvalue problem for graphs on six vertices

TL;DR

This paper advances the understanding of the ordered multiplicity inverse eigenvalue problem for graphs on six vertices, providing a complete solution and exploring spectral attainability for certain bipartite graphs.

Contribution

It offers a complete solution to the ordered multiplicity IEPG for connected graphs of six vertices and analyzes spectral attainability for complete bipartite graphs.

Findings

Complete solution to the ordered multiplicity IEPG for six-vertex graphs.

Identifies spectral limitations for $K_{m,n}$ with $ ext{min}(m,n) extgreater 2$.

Shows certain multiplicity lists are unattainable with arbitrary spectra.

Abstract

For a graph $G$, we associate a family of real symmetric matrices, $\mathcal{S}(G)$, where for any $M \in \mathcal{S}(G)$, the location of the nonzero off-diagonal entries of $M$ are governed by the adjacency structure of $G$. The ordered multiplicity Inverse Eigenvalue Problem of a Graph (IEPG) is concerned with finding all attainable ordered lists of eigenvalue multiplicities for matrices in $\mathcal{S}(G)$. For connected graphs of order six, we offer significant progress on the IEPG, as well as a complete solution to the ordered multiplicity IEPG. We also show that while $K_{m,n}$ with $\min(m,n)\ge 3$ attains a particular ordered multiplicity list, it cannot do so with arbitrary spectrum.