Abstract
The Wiener polarity index Wp(G) of a graph G is the number of unordered pairs of vertices {u, v} of G such that the distance between u and v is 3. In this paper, we study the Nordhaus–Gaddum-type inequality for the Wiener polarity index of a graph G of order n. Due to concerns that both Wp(G) and Wp(G¯) are nontrivial only when diam(G)=3 and diam(G¯)=3, we firstly consider the crucial case and get that 2≤Wp(G)+Wp(G¯)≤⌈n2⌉⌊n2⌋−n+2. Moreover, the bounds are best possible, and the corresponding extremal graphs are also presented. Then we generalize the results to all connected simple graphs. Furthermore, we discuss the Nordhaus–Gaddum-type inequality for trees and unicyclic graphs, respectively.
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