Graphs whose adjacency matrices have rank equal to the number of distinct nonzero rows

For a simple graph G, let rank(G) and dnzr(G) denote respectively the rank and the number of distinct nonzero rows of the adjacency matrix A(G) of G. Equivalent conditions are given for the join G1∨G2 of two vertex-disjoint graphs G1,G2 to satisfy rank(G1∨G2)=dnzr(G1∨G2). A new proof is provided for the known relation rank(G) = dnzr(G) for cographs G. Our approach relies on the concepts of neighborhood equivalence classes, reduced graph and reduced adjacency matrix, and also on a known result that relates the spectrum of the adjacency matrix of a graph with that of its reduced adjacency matrix as well as a new characterization of the nonsingularity of a real symmetric matrix in a special 2 × 2 block form. Our treatment provides ways to construct graphs G, other than cographs, that satisfy rank(G) = dnzr(G). As a side result we also show that every rational number is equal to the sum of the entries of the inverse of the adjacency matrix of a connected nonsingular graph.