Randić energy of digraphs

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