On the Wiener polarity index of graphs

The Wiener polarity index Wp(G) of a graph G is the number of unordered pairs of vertices {u, v} in G such that the distance between u and v is equal to 3. Very recently, Zhang and Hu studied the Wiener polarity index in [Y. Zhang, Y. Hu, 2016] [38]. In this short paper, we establish an upper bound on the Wiener polarity index in terms of Hosoya index and characterize the corresponding extremal graphs. Moreover, we obtain Nordhaus–Gaddum-type results for Wp(G  ). Our lower bound on Wp(G)+Wp(G¯) is always better than the previous lower bound given by Zhang and Hu.