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arXiv · 2307.00376

The Spark of Symmetric Matrices Described by a Graph

Louis Deaett·Shaun Fallat·Veronika Furst·John Hutchens·Lon Mitchell·Yaqi Zhang
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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.

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Louis Deaett, Shaun Fallat, Veronika Furst, John Hutchens, Lon Mitchell, Yaqi Zhang. 2023-07-01. The Spark of Symmetric Matrices Described by a Graph. https://arxiv.org/abs/2307.00376

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