A variant on the graph parameters of Colin de Verdiere: Implications to the minimum rank of graphs

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Francesco Barioli
Shaun Fallat
Leslie Hogben


For a given undirected graph G, the minimum rankof G is defined to be the smallest
possible rankover all real symmetric matrices A whose (i, j)th entry is nonzero whenever i≠ j and
{i, j} is an edge in G. Building upon recent workinvolving maximal coranks (or nullities) of certain
symmetric matrices associated with a graph, a new parameter ξ is introduced that is based on the corankof a different but related class of symmetric matrices. For this new parameter some properties analogous to the ones possessed by the existing parameters are verified. In addition, an attempt is made to apply these properties associated with ξ to learn more about the minimum rankof graphs– the original motivation.

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