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The inverse degree index of a graph G=(V,E) without isolated vertices is defined as ID(G)=ΣvЄv1/dv, where dv is the degree of the vertex v in G. In this paper, we show a relation between the inverse degree of a graph and the inverse degree indices of the primary subgraphs obtained by a general decomposition of G, we establish some relations between the inverse degree index and other known indices and an application to a specific chemical structure is given. [ABSTRACT FROM AUTHOR]

Let G = (V,E) be a graph; a set S ⊆ V is a total k-dominating set if every vertex v ∍ V has at least k neighbors in S. The total k-domination number γkt(G) is the minimum cardinality among all total k-dominating sets. In this paper we obtain several tight bounds for the total k-domination number of a graph. In particular, we investigate the relationship between the total k-domination number of a graph and the order, the size, the girth, the minimum and maximum degree, the diameter, and other domination parameters of the graph. [ABSTRACT FROM AUTHOR]