Strong Domination Index in Fuzzy Graphs

TL;DR

This paper introduces a new topological index called the strong domination index in fuzzy graphs, based on the weight of strong edges, with applications and algorithms for various graph types.

Contribution

It defines the strong domination index and related parameters in fuzzy graphs, providing new tools and inequalities for analyzing fuzzy graph structures.

Findings

Defined the strong domination index (SDI) for fuzzy graphs.

Derived inequalities involving SDI and SDD.

Applied the concepts to various fuzzy graph classes and operations.

Abstract

Topological indices play a vital role in the area of graph theory and fuzzy graph (FG) theory. It has wide applications in the areas such as chemical graph theory, mathematical chemistry, etc. Topological indices produce a numerical parameter associated with a graph. Numerous topological indices are studied due to its applications in various fields. In this article a novel idea of domination index in a FG is defined using weight of strong edges. The strong domination degree (SDD) of a vertex u is defined using the weight of minimal strong dominating set (MSDS) containing u. Idea of upper strong domination number, strong irredundance number, strong upper irredundance number, strong independent domination number, and strong independence number are explained and illustrated subsequently. Strong domination index (SDI) of a FG is defined using the SDD of each vertex. The concept is applied on various FGs like complete FG, complete bipartite and r-partite FG, fuzzy tree, fuzzy cycle and fuzzy stars. Inequalities involving the SDD and SDI are obtained. The union and join of FG is also considered in the study. Applications for SDD of a vertex is provided in later sections. An algorithm to obtain a MSDS containing a particular vertex is also discussed in the article.