Minimum rank of a graph

In mathematics, the minimum rank is a graph parameter formula_1 for a graph "G". It was motivated by the Colin de Verdière graph invariant.
Definition.
The adjacency matrix of an undirected graph is a symmetric matrix whose rows and columns both correspond to the vertices of the graph. Its elements are all 0 or 1, and the element in row "i" and column "j" is nonzero whenever vertex "i" is adjacent to vertex "j" in the graph. More generally, a "generalized adjacency matrix" is any symmetric matrix of real numbers with the same pattern of nonzeros off the diagonal (the diagonal elements may be any real numbers). The minimum rank of formula_2 is defined as the smallest rank of any generalized adjacency matrix of the graph; it is denoted by formula_3.
Properties.
Here are some elementary properties.
Characterization of known graph families.
Several families of graphs may be characterized in terms of their minimum ranks.