Summary: 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: \[\begin{aligned} WW_e(G)&=\sum_{\{f,g\}\subseteq E(G)} (d_e(f,g|G)+d_e^2(f,g|G))\\&=\frac{1}{2}\sum_{f\in E(G)} (d_e(f|G)+d^2_e(f|G)), \end{aligned}\] where \(d_e(f,g|G)\) denotes the distance between the edges \(f=xy\) and \(g=uv\) in \(E(G)\) and \(d_e(f|G)=\sum_{g\in E(G)} d_e(f,g|G)\). 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 classes of 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. Our aim in last section is to investigate products of two or more graphs, and compute edge-hyper Wiener number of some classes of graphs.