Wiener polarity index of fullerenes and hexagonal systems

The Wiener polarity index Wp(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 dG(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, Wp(G)=M2(G)−M1(G)−5Np−3Nh+|E(G)|, where M1(G), M2(G), Np and Nh 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 (9n−60)/2. The extremal values of catacondensed hexagonal systems, hexagonal cacti and polyphenylene chains with respect to the Wiener polarity index are also computed.