Randić energy of digraphs

We assume that D is a directed graph with vertex set [Formula: see text] and arc set [Formula: see text]. A VDB topological index φ of D is defined as [Formula: see text] where [Formula: see text] and [Formula: see text] denote the outdegree and indegree of vertices u and v, respectively, and [Formula: see text] is a bivariate symmetric function defined on nonnegative real numbers. Let [Formula: see text] be the [Formula: see text] general adjacency matrix defined as [Formula: see text] if [Formula: see text] , and 0 otherwise. The energy of D with respect to a VDB index φ is defined as [Formula: see text] , where [Formula: see text] are the singular values of the matrix [Formula: see text]. We will show that in case [Formula: see text] is the Randić index, the spectral norm of [Formula: see text] is equal to 1, and rank of [Formula: see text] is equal to rank of the adjacency matrix of D. Immediately after, we illustrate by means of examples, that these properties do not hold for most well-known VDB topological indices. Taking advantage of nice properties the Randić matrix has, we derive new upper and lower bounds for the Randić energy [Formula: see text] in digraphs. Some of these generalize known results for the Randić energy of graphs. Also, we deduce a new upper bound for the Randić energy of graphs in terms of rank, concretely, we show that [Formula: see text] for all graphs G, and equality holds if and only if G is a disjoint union of complete bipartite graphs.