Let G be a connected graph with the vertex set V = {v 1 , v 2 , . . ., v n }, where n ≥ 2. Denote by d i the degree of the vertex v i for i = 1, 2, . . . , n. If v i and v j are adjacent in G, we write i ∼ j, otherwise we write i j. The variable sum exdeg index and coindex of G are defined as, respectively, where 'a' is a positive real number different from 1. Some inequalities involving SEI a (G) or/and SEI a (G) are derived. Special cases of the obtained inequalities are also discussed for unicyclic graphs.

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In this paper, the notion of energy extended to spherical fuzzy graph. The adjacency matrix of a spherical fuzzy graph is defined and we compute the energy of a spherical fuzzy graph as the sum of absolute values of eigenvalues of the adjacency matrix of the spherical fuzzy graph. Also, the lower and upper bounds for the energy of spherical fuzzy graphs are obtained.

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Abstract:In this paper, effective adjacency matrix of a fuzzy graph is introduced. An algorithm for finding a minimal complementary nil dominating set of fuzzy graph is studied with suitable examples.