ELA, Volume 13, pp. 386-404, December 2005, abstract.

A Variant on the Graph Parameters of Colin de Verdiere:
Implications to the Minimum Rank of Graphs

Francesco Barioli, Shaun Fallat, and Leslie Hogben

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 does not equal 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 xi 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 xi to learn
more about the minimum rank of graphs - the original motivation.