Open problems on Euler-Sombor index of chemical graphs

Abstract

The Euler-Sombor index is a vertex-degree-based topological index defined for a graph G = (V,E) as where d_u and d_v denote the degrees of adjacent vertices u and v. In prior work [6], the chemical trees and unicyclic graphs with the minimum Euler-Sombor indices were characterized, prompting the search for extremal values over all chemical graphs. This paper extends that work by determining the first three minimum Euler-Sombor indices among chemical bicyclic graphs and the first seven among chemical tricyclic graphs. Furthermore, we prove that the minimum Euler-Sombor index for both unicyclic and bicyclic chemical graphs is attained when the maximum degree is at most 4, thereby conclusively establishing the minimum values for these graph classes.