Eccentric Connectivity Index of the Non-Commuting Graph Associated to the Dihedral Groups of Order at Most 12

Abstract

A topological index is a numerical value or invariant in mathematics that characterizes specific topological aspects of a space, manifold, or mathematical object. Topological indices are used to differentiate between topological spaces or to capture specific characteristics of their structure. Meanwhile, a non-commuting graph is a graph in which two unique vertices are adjacent if and only if they do not commute, that is , and it consists of the non-central elements set in a group as a vertex. In this paper, Maple software constructed the non-commuting graph of the dihedral groups of order at most 12. Then, the degree and distance of the non-commuting graph for dihedral groups are found. The eccentric connectivity index of the non-commuting graph of dihedral groups of order at most 12 is computed using its definition. As a result, the eccentric connectivity index of non-commuting graphs for dihedral groups increases as the order of the groups increases. In real life, one of the eccentric connectivity index's effects is that it can be utilized as a chemical descriptor in drug discovery to predict biological activities such as binding affinities to target proteins or enzymes.