Revised and edge revised Szeged indices of graphs

The revised Szeged index is a molecular structure descriptor equal to the sum of products [ n u ( e ) + n 0 ( e ) / 2] × [ n v ( e ) + n 0 ( e ) / 2] over all edges e = u v of the molecular graph G , where n 0 ( e ) is the number of vertices equidistant from u and v , n u ( e ) is the number of vertices whose distance to vertex u is smaller than the distance to vertex v and n v ( e ) is defined analogously. In this paper, new formula for computing this molecular descriptor is presented by which it is possible to reprove most of results given in [M. Aouchiche and P. Hansen, On a conjecture about the Szeged index, European J. Combin. 31 (2010), 1662–1666]. We also present an edge version of this graph invariant. At the end of the paper an open question is presented.