more on edge hyper wiener index of graphs

‎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: ‎$$ww_{e}(g)=sum_{{f,g}subseteq e(g)}(d_{e}(f,g|g)+d_{e}^{2}(f,g|g))=frac{1}{2}sum_{fin e(g)}(d_{e}(f|g)+d^{2}_{e}(f|g)),$$‎ ‎where $d_{e}(f,g|g)$ denotes the distance between the edges $f=xy$ and $g=uv$ in $e(g)$ and $d_{e}(f|g)=sum_{gin e(g)}d_{e}(f,g|g)$‎. ‎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.