The paper discusses the edge hyper-Zagreb index of a graph, which is calculated by replacing vertex degrees with edge degrees. The degree of an edge is determined by adding up the degrees of the end vertices of the edge and subtracting 2. We examine the edge hyper-Zagreb index of the Cartesian product and join of graphs, and also calculate it for organic linear Acenes molecules with the formula (C4n+2H2n+4). We establish a correlation between topological indices based on the number of rings and predict thermodynamic properties of Acenes family, such as electron affinity, bond, heat of formation and gap energy, using the Topological Indices Method (T IM).