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The concepts of domination and topological index hold great significance within the realm of graph theory. Therefore, it is pertinent to merge these concepts to derive the domination index of a graph. A novel concept of the domination index is introduced, which utilizes the domination degree of a vertex. The domination degree of a vertex Formula: see text is defined as the minimum cardinality of a minimal dominating set (MDS) that includes Formula: see text Methods to find a MDS containing a particular vertex is also discussed in the study. The notion of domination degree and domination index are studied for graphs like complete graphs, complete bipartite, Formula: see text partite graphs, cycles, wheels, paths, book graphs, windmill graphs, Kragujevac trees. The study is extended to operation in graphs. Inequalities involving domination degree and already established graph parameters are discussed. An application of domination degree is discussed in facility allocation in a city.
Nair et al. (2024) studied this question.
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