Kemeny's constant and Wiener index on trees

TL;DR

This paper establishes a direct relation between Kemeny's constant and Wiener index on trees, providing new formulas, simplifying proofs of extremal properties, and exploring co-Kemeny's mates with identical Kemeny's constants.

Contribution

It introduces a new formula linking Kemeny's constant and Wiener index on trees, and explores properties of graphs with identical Kemeny's constants.

Findings

Derived a new formula for Kemeny's constant from Wiener index

Simplified proofs of extremal tree properties using the relation

Identified conditions for trees to have maximum Kemeny's constant

Abstract

On trees of fixed order, we show a direct relation between Kemeny's constant and Wiener index, and provide a new formula of Kemeny's constant from the relation with a combinatorial interpretation. Moreover, the relation simplifies proofs of several known results for extremal trees in terms of Kemeny's constant for random walks on trees. Finally, we provide various families of co-Kemeny's mates, which are two non-isomorphic connected graphs with the same Kemeny's constant, and we also give a necessary condition for a tree to attain maximum Kemeny's constant for trees with fixed diameter.