Shahrood University of TechnologyJournal of Algebraic Systems2345-51284220170101MORE ON EDGE HYPER WIENER INDEX OF GRAPHS13515385410.22044/jas.2017.854ENA. AlhevazDepartment of Mathematics, Shahrood University of Technology, P.O. Box: 316-
3619995161, Shahrood, Iran.0000-0001-6167-607XM. BaghipurDepartment of Mathematics, Shahrood University of Technology, P.O. Box: 316-
3619995161, Shahrood, Iran.0000-0002-9069-9243Journal Article20160410Let G=(V(G),E(G)) be a simple connected graph with vertex set V(G) and edge<br /> set E(G). The (first) edge-hyper Wiener index of the graph G is defined as:<br /> $$WW_{e}(G)=\sum_{\{f,g\}\subseteq E(G)}(d_{e}(f,g|G)+d_{e}^{2}(f,g|G))=\frac{1}{2}\sum_{f\in E(G)}(d_{e}(f|G)+d^{2}_{e}(f|G)),$$<br /> where d<sub>e</sub>(f,g|G) denotes the distance between the edges f=xy and g=uv in E(G) and d<sub>e</sub>(f|G)=∑<sub><span style="font-size: 8.33333px;">g€(G)</span></sub><span style="font-size: 8.33333px;">d<sub>e</sub>(f,g|G).</span><br /> In this paper we use a method, which applies group theory to graph theory, to improving<br /> mathematically computation of the (first) edge-hyper Wiener index in certain graphs.<br /> 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.<br /> 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.https://jas.shahroodut.ac.ir/article_854_2486403d0b8da2a0bb248f7cd1fcd96b.pdf