This paper is aimed at presenting the unified integral operator in its generalized form utilizing the unified Mittag-Leffler function in its kernel. We prove the boundedness of this newly defined operator. A fractional integral operator comprising a unified Mittag-Leffler function is used to establish further Minkowski-type integral inequalities. Several related fractional integral inequalities that have recently been published in various articles can be inferred.

Topological indices are the numbers associated with the graphs of chemical compounds/networks that help us to understand their properties. The aim of this paper is to compute topological indices for the hierarchical hypercube networks. We computed Hosoya polynomials, Harary polynomials, Wiener index, modified Wiener index, hyper-Wiener index, Harary index, generalized Harary index, and multiplicative Wiener index for hierarchical hypercube networks. Our results can help to understand topology of hierarchical hypercube networks and are useful to enhance the ability of these networks. Our results can also be used to solve integral equations.