Sombor index of directed graphs

Area of Science:

• Graph Theory
• Discrete Mathematics
• Chemical Informatics

Background:

• The Sombor index, a molecular descriptor, has shown predictive potential in graph theory.
• Its application to digraphs (directed graphs) has not been previously explored.

Purpose of the Study:

• To initiate the study of the Sombor index for digraphs.
• To establish sharp upper and lower bounds for the Sombor index in various classes of digraphs.

Main Methods:

• Definition of the Sombor index for digraphs using out-degrees and in-degrees.
• Mathematical analysis to derive bounds for specific digraph classes.

Main Results:

• The Sombor index is defined for digraphs as the sum over all arcs (u,v) of (out-degree(u) + in-degree(v)).
• Sharp upper and lower bounds are determined for digraphs with non-isolated vertices, connected digraphs, strongly connected digraphs, oriented trees, and orientations of a fixed graph.

Conclusions:

• The Sombor index is a valuable descriptor for digraphs, analogous to its utility in graphs.
• The established bounds provide a foundation for future research in the spectral properties of digraphs.