The minimum rank of matrices and the equivalence class graph

For a given connected (undirected) graph G, the minimum rank of G = (V(G), E(G)) is defined to be the smallest possible rank over all hermitian matrices A whose (i, j)th entry is non-zero if and only if i not equal j and {i, j} is an edge in G ({i, j} is an element of E (G)). For each vertex x in G (x is an element of V (G)), N (x) is the set of all neighbors of x. Let R be the equivalence relation on V (G) such that for all(x),V-y is an element of(G) xRy double left right arrow N(x) = N(y). Our aim is find classes of connected graphs G = (V(G), E(G)), such that the minimum rank of G is equal to the number of equivalence classes for the relation R on V (G). (C) 2007 Elsevier Inc. All rights reserved.