Inertia Sets For Families of Graphs

TL;DR

This paper explores inertia tables of graphs, which display eigenvalue sign distributions, providing new formulas and improved notation to advance understanding of the inverse eigenvalue problem and minimum rank problem in graph theory.

Contribution

It introduces new formulas for inertia tables of simple graphs and enhances notation, aiding the study of inverse eigenvalue problems in graph theory.

Findings

New general formulas for inertia tables of graphs

Improved notation for inertia tables

Enhanced tools for inverse eigenvalue problem

Abstract

This paper consists of a few results, discovered and proved during the 2012-2013 research group at Eastern Oregon University. Inertia tables are a visual representation of the possible inertias of a given graph. The inertia of a graph counts the number of real positive and negative eigenvalues of its corresponding adjacency matrix. The problem of studying inertia tables is directly related to the inverse eigenvalue problem and can be used as a tool for the minimum rank problem. This paper describes the inverse eigenvalue problem, and tools used. We describe a number of new general formulas for various simple undirected graphs and improved upon an established notation for inertia tables.