Graph Invertibility and Median Eigenvalues

TL;DR

This paper characterizes the inverse of weighted graphs using Sachs subgraphs, enabling bounds on median eigenvalues relevant to quantum chemistry, and applies these results to specific graph classes like stellated and corona graphs.

Contribution

It provides a novel combinatorial characterization of graph inverses based on Sachs subgraphs, facilitating eigenvalue analysis without matrix computations.

Findings

Graphs with a unique Sachs subgraph and a perfect matching have spectra split about the origin.

Median eigenvalues of stellated graphs of trees are in different halves of [-1,1].

Median eigenvalues of corona graphs are in different halves of [-1,1].

Abstract

Let $(G,w)$ be a weighted graph with a weight-function $w: E(G)\to \mathbb R\backslash\{0\}$. A weighted graph $(G,w)$ is invertible to a new weighted graph if its adjacency matrix is invertible. A graph inverse has combinatorial interest and can be applied to bound median eigenvalues of a graph such as have physical meanings in Quatumn Chemistry. In this paper, we characterize the inverse of a weighted graph based on its Sachs subgraphs that are spanning subgraphs with only $K_2$ or cycles (or loops) as components. The characterization can be used to find the inverse of a weighted graph based on its structures instead of its adjacency matrix. If a graph has its spectra split about the origin, i.e., half of eigenvalues are positive and half of them are negative, then its median eigenvalues can be bounded by estimating the largest and smallest eigenvalues of its inverse. We characterize graphs with a unique Sachs subgraph and prove that these graphs has their spectra split about the origin if they have a perfect matching. As applications, we show that the median eigenvalues of stellated graphs of trees and corona graphs belong to different halves of the interval $[-1,1]$.