Inverse eigenvalue problem for Laplacian matrices of a graph

TL;DR

This paper investigates the inverse eigenvalue problem for Laplacian matrices of graphs, exploring possible spectra and multiplicities for various graph families, and introduces a new approach to minimizing variance in weighted Laplacians.

Contribution

It provides new theoretical and numerical insights into the spectra of generalized Laplacian matrices for specific graph families and introduces a novel variance minimization method.

Findings

Characterization of realizable spectra for certain graph families.

Analysis of multiplicity lists for stars and complete graphs.

Development of a variance minimization technique for weighted Laplacians.

Abstract

For a given graph $G$, we aim to determine the possible realizable spectra for a generalized (or sometimes referred to as a weighted) Laplacian matrix associated with $G$. This new specialized inverse eigenvalue problem is considered for certain families of graphs and graphs on a small number of vertices. Related considerations include studying the possible ordered multiplicity lists associated with stars and complete graphs and graphs with a few vertices. Finally, we present a novel investigation, both theoretically and numerically, the minimum variance over a family of generalized Laplacian matrices with a size-normalized weighting.