Document Type: Original Manuscript

Authors

Department of Mathematics, Shahrood University of Technology, P.O. Box: 316- 3619995161, Shahrood, Iran.

Abstract

‎Let G=(V(G),E(G)) be a simple connected graph with vertex set V(G) and edge‎
‎set E(G)‎. ‎The (first) edge-hyper Wiener index of the graph G is defined as‎:
‎$$WW_{e}(G)=\sum_{\{f,g\}\subseteq E(G)}(d_{e}(f,g|G)+d_{e}^{2}(f,g|G))=\frac{1}{2}\sum_{f\in E(G)}(d_{e}(f|G)+d^{2}_{e}(f|G)),$$‎
‎where de(f,g|G) denotes the distance between the edges f=xy and g=uv in E(G) and de(f|G)=∑g€(G)de(f,g|G).
‎In this paper we use a method‎, ‎which applies group theory to graph theory‎, ‎to improving‎
‎mathematically computation of the (first) edge-hyper Wiener index in certain graphs‎.
‎We give also upper and lower bounds for the (first) edge-hyper Wiener index of a graph in terms of its size and Gutman index‎. ‎Also we investigate products of two or more graphs and compute the second edge-hyper Wiener index of the some classes of graphs‎.
‎Our aim in last section is to find a relation between the third edge-hyper Wiener index of a general graph and the hyper Wiener index of its line graph‎. of two or more graphs and compute edge-hyper Wiener number of some classes of graphs‎.

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