The minimum rank of matrices and the equivalence class graph Original Research Article

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≠j and {i,j} is an edge in G ({i,j}set membership, variantE(G)). For each vertex x in G (xset membership, variantV(G)), N(x) is the set of all neighbors of x. Let R be the equivalence relation on V(G) such thatimageOur 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).