The Spark of Symmetric Matrices Described by a Graph

TL;DR

This paper explores the relationship between the sparsity of null vectors in symmetric matrices and the structural properties of the underlying graph, introducing the concept of graph spark and connecting it to various graph invariants.

Contribution

It introduces the notion of graph spark based on matrix null vectors and links it to established graph concepts like minimum rank and vertex connectivity.

Findings

Defined graph spark as the minimal spark over matrices with a given graph pattern.

Connected graph spark to minimum rank, forts, and orthogonal representations.

Provided insights into how graph structure influences matrix null space properties.

Abstract

We investigate the sparsity of null vectors of real symmetric matrices whose off-diagonal pattern of zero and nonzero entries is described by the adjacencies of a graph. We use the definition of the spark of a matrix, the smallest number of nonzero coordinates of any null vector, to define the spark of a graph as the smallest possible spark of a corresponding matrix. We study connections of graph spark to well-known concepts including minimum rank, forts, orthogonal representations, Parter and Fiedler vertices, and vertex connectivity.