The modified Schultz index of graph operations

Carvalho, Paula and Rama, Paula

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creative_2017_26_1_53_60

DOI: https://doi.org/10.37193/CMI.2017.01.07

Issue no: Vol 26/2017 no. 1 Tags: Graph theory, graph operations, topological indices, modified Schultz index

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Given a simple and connected graph $G$ with vertex set $V$ , denoting by $d_G(u)$ , the degree of a vertex $u$ and by $d_G(u,v)$ the distance of two vertices, the modified Schultz index of $G$ is given by $S^*(G)= \sum_{\{u,v\} \subseteq V} d_G(u) \, d_G(v)\, d_G(u,v),$ where the summation goes over all non ordered pairs of vertices of $G$ .
In this paper we consider some graph operations, namely cartesian product, complete product, composition and subdivision, and we obtain explicit formulae for the modified Schultz index of a graph in terms of the number of vertices and edges as well as some other topological invariants such as the Wiener index, the Schultz index and the first and second Zagreb indices.