The inverse inertia problem for graphs

TL;DR

This paper investigates the inverse inertia problem for graphs, providing complete solutions for trees, formulas for graphs with cut vertices, and exploring restrictions on inertia sets, advancing understanding of graph-related matrix inertias.

Contribution

It introduces the maximal disconnection numbers as a new graph parameter and offers formulas for inertia sets of graphs with cut vertices, addressing the inverse inertia problem comprehensively.

Findings

Complete solution for trees using maximal disconnection numbers

Formula for inertia sets of graphs with cut vertices

Identification of graphs that are not inertia-balanced

Abstract

Let G be an undirected graph on n vertices and let S(G) be the set of all real symmetric n x n matrices whose nonzero off-diagonal entries occur in exactly the positions corresponding to the edges of G. The inverse inertia problem for G asks which inertias can be attained by a matrix in S(G). We give a complete answer to this question for trees in terms of a new family of graph parameters, the maximal disconnection numbers of a graph. We also give a formula for the inertia set of a graph with a cut vertex in terms of inertia sets of proper subgraphs. Finally, we give an example of a graph that is not inertia-balanced, and investigate restrictions on the inertia set of any graph.