  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).
