| The study of topological indices and vulnerability parameters of graphs is a very important part of graph theory. A topological index is a real number related to a graph. It must be a structural invariant, i.e., it preserves by every graph automorphisms. The Wiener index W is the first topological index to be used in chemistry. There are several topological indices such as, the Szeged index Sz, the Randic index R, the Hosoya index Z, the Merrifield-Simmons index a, and the vertex and edge Padmakar-Ivan index PIv and PIe. These topological indices have found applications as means to model chemical, pharmaceutical and other properties of molecules.A communication network can be modeled by a connected graph whose vertices repre-sent the stations and whose edges represent the lines of communication. There are several measures of the vulnerability of a network. When measuring the vulnerability of a net-work, we consider its corresponding graph. The vulnerability parameters one generally encounters are connectivity and edge-connectivity, which measure the vulnerability of a graph, but they do not take into account what remains after destruction. To measure the vulnerability of networks more properly, some vulnerability parameters have been intro-duced and studied. Among them are toughness, scattering number, integrity and rupture degree, each of which measures not only the difficulty of breaking down the network but also the effect of the damage. In general, for most of the aforementioned parameters, the corresponding computing problem is NP-hard. So it is of interest to give formulae or algorithms for computing these parameters for special classes of graphs.In this paper, we study Wiener, hyper-Wiener and vertex PI indices of the Kronecker product G×Kn of a connected graph G and a complete graph Kn for n≥3, and determine vertex-neighbor-scattering number, vertex-neighbor-integrity and rupture degree of the Kronecker product of complete graphs Km x Kn for n≥m≥2 and n≥3.
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