On the inverse eigenvalue problem for block graphs

TL;DR

This paper completely solves the inverse eigenvalue problem for certain block graphs, including lollipop and barbell graphs, by introducing a new technique that constructs matrices with desired spectral properties.

Contribution

It introduces a novel method using the strong spectral property to realize spectra for block graphs, expanding understanding of their inverse eigenvalue problem.

Findings

Complete solution for clique-path graphs, including lollipop and barbell graphs.

New technique to construct matrices with prescribed spectra by appending cliques.

Many spectra are realizable for block graphs using the proposed method.

Abstract

The inverse eigenvalue problem of a graph $G$ aims to find all possible spectra for matrices whose $(i,j)$-entry, for $i\neq j$, is nonzero precisely when $i$ is adjacent to $j$. In this work, the inverse eigenvalue problem is completely solved for a subfamily of clique-path graphs, in particular for lollipop graphs and generalized barbell graphs. For a matrix $A$ with associated graph $G$, a new technique utilizing the strong spectral property is introduced, allowing us to construct a matrix $A'$ whose graph is obtained from $G$ by appending a clique while arbitrary list of eigenvalues is added to the spectrum. Consequently, many spectra are shown realizable for block graphs.