The Second Omega Index

Abstract

Background: The omega index has been recently introduced to identify a variety of topological and combinatorial aspects of a graph with a new viewpoint. As a continuing study of the omega index, by considering the incidence of edges and vertices to the adjacency of the vertices, in this paper, we have introduced the second omega index Ω2 and then computed it over some well-known graph classes. Methods: Many combinatorial counting methods have been utilized in the proofs. The edge partition is frequently used throughout the work. Naturally, some graph theoretical lemmas are also used. Results: In particular, trees having the smallest and largest Ω2 have been constructed. Finally, the second omega index of some derived graphs, such as line graphs, subdivision graphs, and vertex-semitotal graphs, has been presented. Conclusion: Omega invariant has already been explored in many papers. It has been defined in terms of vertex degrees. Vertices correspond to the atoms in a molecule and a calculation, which only depends on the atomic parameters, is not even comparable with a calculation containing both atoms and chemical bonds between them. With this idea in mind, we have evaluated some mathematical properties of the second omega index, which has great potential in chemical applications related to the number of cycles in the molecular graph.