Graphs with a given diameter that maximise the Wiener index

Abstract

The Wiener index of a graph is one of the most recognised and very well-researched topological indices, i.e. graph theoretic invariants of molecular graphs. Nonetheless, some interesting questions remain largely unsolved despite being easy to state and comprehend. In this paper, we investigate a long-standing question raised by Plesník in 1984, namely, which graphs with a given diameter d attain the maximum value with respect to the Wiener index. Our approach to the problem is twofold – first we investigate the graphs with diameter smaller than or equal to 4, and then restrict our attention to graphs with diameter equal to n−c for c ≥ 1. Specifically, we provide a complete characterisation of sought-after graphs for 1 ≤ c ≤ 4 and solve the general case for c small enough in comparison to n. Along the way, we state some conjectures and propose an extension to our work.