The Edge Version of the Szeged Index

The Szeged index is a molecular structure descriptor equal to the sum of productsover all edges e = (uv) of the molecular graph G, where n u (e | G) is the number of vertices whose distance to vertex u is smaller than the distance to vertex v, and where n v (e | G) is defined analogously.We now examine the edge version of the Szeged index: the sum of products m u (e | G) ⋅ m v (e | G) over all edges e = (uv) of the molecular graph G, where m u (e | G) is the number of edges whose distance to vertex u is smaller than the distance to vertex v, and where m v (e | G) is defined analogously.The basic properties of this novel structure descriptor are established.Most of these are analogous to the properties of the ordinary Szeged index, but some remarkable differences are also recognized.