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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.

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]

Some years ago, the harmonic polynomial was introduced to study the harmonic topological index. Here, using this polynomial, we obtain several properties of the harmonic index of many classical symmetric operations of graphs: Cartesian product, corona product, join, Cartesian sum and lexicographic product. Some upper and lower bounds for the harmonic indices of these operations of graphs, in terms of related indices, are derived from known bounds on the integral of a product on nonnegative convex functions. Besides, we provide an algorithm that computes the harmonic polynomial with complexity O (n 2) . [ABSTRACT FROM AUTHOR]