Sombor index of directed graphs

Let D be a digraph with set of arcs A. The Sombor index of D is defined as where $$ and $$ are the out-degree and in-degree of the vertices u and v of D. When D is a graph, we recover the Sombor index of graphs, a molecular descriptor recently introduced with a good predictive potential and a great research activity this year. In this paper we initiate the study of the Sombor index of digraphs. Specifically, we find sharp upper and lower bounds for $$ over the class $$ of digraphs with n non-isolated vertices, the classes $$ and $$ of connected and strongly connected digraphs on n vertices, respectively, the class of oriented trees $$ with n vertices, and the class $$ of orientations of a fixed graph G.