 
Summary: ELA
A VARIANT ON THE GRAPH PARAMETERS OF COLIN DE
VERDI`ERE: IMPLICATIONS TO THE MINIMUM RANK OF
GRAPHS
FRANCESCO BARIOLI, SHAUN FALLAT, AND LESLIE HOGBEN§
Abstract. For a given undirected graph G, the minimum rank of G is defined to be the smallest
possible rank over 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 work involving maximal coranks (or nullities) of certain
symmetric matrices associated with a graph, a new parameter is introduced that is based on the
corank of 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 to learn more about the minimum rank of graphs
the original motivation.
Key words. Graphs, Minimum rank, Graph minor, Corank, Strong Arnold property, Symmetric
matrices.
AMS subject classifications. 15A18, 05C50.
1. Introduction. Recent work and subsequent results have fueled interest in
important areas such as spectral graph theory and certain types of inverse eigenvalue
problems. Of particular interest here is to bring together some of the pioneering work
of Y. Colin de Verdi`ere (specifically his parameter related to planarity of graphs) and
