For a given undirected graph G, the minimum rankof G is defined to be the smallest possible rank over all real symmetric matrices A whose (i, j)th entry is nonzero whenever i 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 rankof graphs - the original motivation.