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Transitivity on graphs is a concept widely investigated. This suggest to analyze the action of automorphisms on other sets. In this paper, we study the action on the family of γ -sets (minimum dominating sets), the graph is called γ -transitive if given two γ -sets there exists an automorphism which maps one onto the other. We deal with two families: paths P n and cycles C n . Their γ -sets are fully characterized and the action of the automorphism group on the family of γ -sets is fully analyzed. [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]