MORE ON EDGE HYPER WIENER INDEX OF GRAPHS

ýLet $G=(V(G),E(G))$ be a simple connected graph with vertex set $V(G)$ and edgeý ýset $E(G)$ý. ýThe (first) edge-hyper Wiener index of the graph $G$ is defined asý: ý$$WW_{e}(G)=\sum_{\{f,g\}\subseteq E(G)}(d_{e}(f,g|G)_{e}^{2}(f,g|G))=\frac 1}{2}\sum_{f\in E(G)}(d_{e}(f|G)^{2}_{e}(f|G)),$$ý ýwhere $d_{e}(f,g|G)$ denotes the distance between the edges $f=xy$ and $g=uv$ in $E(G)$ and $d_{e}(f|G)=\sum_{g\in E(G)}d_{e}(f,g|G)$ý. ýIn this paper we use a methodý, ýwhich applies group theory to graph theoryý, ýto improvingý ýmathematically computation of the (first) edge-hyper Wiener index in certain graphsý. ýWe give also upper and lower bounds for the (first) edge-hyper Wiener index of a graph in terms of its size and Gutman indexý. ýAlso we investigate products of two or more graphs and compute the second edge-hyper Wiener index of the some classes of graphsý. ýOur 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 graphý. of two or more graphs and compute edge-hyper Wiener number of some classes of graphsý.