Sombor index of directed graphs

Abstract

Let D be a digraph with set of arcs A. The Sombor index of D is defined as $\mathrm{SO}\left(D\right)=\frac{1}{2}\sum _{uv\in A}\sqrt{{\left(d_u^{+}\right)}^2+{\left(d_v^{-}\right)}^2},$ where $d_u^{+}$ and $d_v^{-}$ 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 $\mathrm{SO}$ over the class $D_n$ of digraphs with n non-isolated vertices, the classes $C_n$ and $S_n$ of connected and strongly connected digraphs on n vertices, respectively, the class of oriented trees $\mathrm{OT}\left(n\right)$ with n vertices, and the class $O\left(G\right)$ of orientations of a fixed graph G.

Keywords: Connected digraphs; Digraphs; Orientations; Sombor index; Strongly connected digraphs.