Wiener polarity index of fullerenes and hexagonal systems

The Wiener polarity index W p ( G ) of a molecular graph G of order n is the number of unordered pairs of vertices u , v of G such that the distance d G ( u , v ) between u and v is 3. In this note, it is proved that in a triangle- and quadrangle-free connected graph G with the property that the cycles of G have at most one common edge, W p ( G ) = M 2 ( G ) − M 1 ( G ) − 5 N p − 3 N h + | E ( G ) | , where M 1 ( G ) , M 2 ( G ) , N p and N h denoted the first Zagreb index, the second Zagreb index, the number of pentagons and the number of hexagons, respectively. As a special case, it is proved that the Wiener polarity index of fullerenes with n carbon atoms is ( 9 n − 60 ) / 2 . The extremal values of catacondensed hexagonal systems, hexagonal cacti and polyphenylene chains with respect to the Wiener polarity index are also computed.