Inequalities Involving the Rank of a Graph

Let \(r(G)\) denote the rank, over the field of rational numbers, of the adjacency matrix of a graph \(G\). Van Nuffelen and Ellingham have obtained several inequalities which relate \(r(G)\) to other graph parameters such as chromatic number, clique number, Dilworth number, and domination number. We obtain additional results of this type. Our main theorem is that for graphs \(G\) having no isolated vertices, \(OIR(G) \leq r(G)\), where \(OIR(G)\) denotes the upper open irredundance number of \(G\).