On the main distance-based entropies: the eccentricity- and Wiener-entropy

TL;DR

This paper introduces the Wiener-entropy and analyzes its properties alongside eccentricity-entropy, providing asymptotic extremal behaviors, resolving three conjectures, and proposing a new conjecture about Wiener-entropy's minimization behavior.

Contribution

It defines the Wiener-entropy, compares it with eccentricity-entropy, and solves three existing conjectures while proposing a new one about Wiener-entropy.

Findings

Wiener-entropy of graphs is more spread than eccentricity-entropy.

Resolved three conjectures on eccentricity-entropy.

Proposed a new conjecture on Wiener-entropy's minimization.

Abstract

We define the Wiener-entropy, which is together with the eccentricity-entropy one of the most natural distance-based graph entropies. By deriving the (asymptotic) extremal behaviour, we conclude that the Wiener-entropy of graphs of a given order is more spread than is the case for the eccentricity-entropy. We solve $3$ conjectures on the eccentricity-entropy and give a conjecture on the Wiener-entropy related to some surprising behaviour on the graph minimizing it.