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Given any function f : Z + → R + , let us define the f-index I f (G) = ∑ u ∈ V (G) f (d u) and the f-polynomial P f (G , x) = ∑ u ∈ V (G) x 1 / f (d u) − 1 , for x > 0 . In addition, we define P f (G , 0) = lim x → 0 + P f (G , x) . We use the f-polynomial of a large family of topological indices in order to study mathematical relations of the inverse degree, the generalized first Zagreb, and the sum lordeg indices, among others. In this paper, using this f-polynomial, we obtain several properties of these indices of some classical graph operations that include corona product and join, line, and Mycielskian, among others. [ABSTRACT FROM AUTHOR]

It is well-known that the different products of graphs are some of the more symmetric classes of graphs. Since we are interested in hyperbolicity, it is interesting to study this property in products of graphs. Some previous works characterize the hyperbolicity of several types of product graphs (Cartesian, strong, join, corona and lexicographic products). However, the problem with the direct product is more complicated. The symmetry of this product allows us to prove that, if the direct product G 1 × G 2 is hyperbolic, then one factor is bounded and the other one is hyperbolic. Besides, we prove that this necessary condition is also sufficient in many cases. In other cases, we find (not so simple) characterizations of hyperbolic direct products. Furthermore, we obtain good bounds, and even formulas in many cases, for the hyperbolicity constant of the direct product of some important graphs (as products of path, cycle and even general bipartite graphs). [ABSTRACT FROM AUTHOR]