Wiener polarity index of fullerenes and hexagonal systems

The Wiener polarity index Wp(G)Wp(G) of a molecular graph GG of order nn is the number of unordered pairs of vertices uu, vv of GG such that the distance dG(u,v)dG(u,v) between uu and vv is 3. In this note, it is proved that in a triangle- and quadrangle-free connected graph GG with the property that the cycles of GG have at most one common edge, Wp(G)=M2(G)−M1(G)−5Np−3Nh+|E(G)|Wp(G)=M2(G)−M1(G)−5Np−3Nh+|E(G)|, where M1(G)M1(G), M2(G)M2(G), NpNp and NhNh 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 nn carbon atoms is (9n−60)/2(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.