Sombor index of directed graphs

Abstract

Let D be a digraph with set of arcs A. The Sombor index of D is defined asSO(D)=12∑uv∈A(du+)2+(dv−)2,where du+ and dv− 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 SO over the class Dn of digraphs with n non-isolated vertices, the classes Cn and Sn of connected and strongly connected digraphs on n vertices, respectively, the class of oriented trees OT(n) with n vertices, and the class O(G) of orientations of a fixed graph G.