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A topic of current interest in the study of topological indices is to find relations between some index and one or several relevant parameters and/or other indices. In this paper we study two general topological indices A&alpha, and B&alpha, defined for each graph H=(V(H),E(H)) by A&alpha, (H)=&sum, ij&isin, E(H)f(di,dj)&alpha, i&isin, V(H)h(di)&alpha, where di denotes the degree of the vertex i and &alpha, is any real number. Many important topological indices can be obtained from A&alpha, by choosing appropriate symmetric functions and values of &alpha, This new framework provides new tools that allow to obtain in a unified way inequalities involving many different topological indices. In particular, we obtain new optimal bounds on the variable Zagreb indices, the variable sum-connectivity index, the variable geometric-arithmetic index and the variable inverse sum indeg index. Thus, our approach provides both new tools for the study of topological indices and new bounds for a large class of topological indices. We obtain several optimal bounds of A&alpha, (respectively, B&alpha, ) involving A&beta, (respectively, B&beta, ). Moreover, we provide several bounds of the variable geometric-arithmetic index in terms of the variable inverse sum indeg index, and two bounds of the variable inverse sum indeg index in terms of the variable second Zagreb and the variable sum-connectivity indices.

The object of this paper is to derive a double integral in terms of the Hurwitz–Lerch zeta function. Almost all Hurwitz–Lerch zeta functions have an asymmetrical zero-distribution. Special cases are evaluated in terms of fundamental constants. All the results in this work are new.