Counterexamples to conjectures on graph distance measures based on topological indexes

TL;DR

This paper disproves three conjectures on graph distance measures derived from topological indices by providing explicit counterexamples involving trees with identical indices but different degree sequences.

Contribution

It introduces specific classes of trees that serve as counterexamples, challenging existing conjectures on relationships between graph distance measures and topological indices.

Findings

Disproved three conjectures on graph distance measures

Constructed trees with same indices but different degree sequences

Provided explicit counterexamples to proposed inequalities

Abstract

In this paper we disprove three conjectures from [M. Dehmer, F. Emmert-Streib, Y. Shi, Interrelations of graph distance measures based on topological indices, PLoS ONE 9 (2014) e94985] on graph distance measures based on topological indices by providing explicit classes of trees that do not satisfy proposed inequalities. The constructions are based on the families of trees that have the same Wiener index, graph energy or Randic index - but different degree sequences.