On the relationship between variable Wiener index and variable Szeged index

TL;DR

This paper investigates the relationship between variable Wiener and Szeged indices in graphs, disproves a strong conjecture, and confirms a weaker conjecture through majorization, advancing understanding of these graph invariants.

Contribution

The paper disprove the strong conjecture and prove the weak conjecture for all graphs, clarifying the relationship between the variable Wiener and Szeged indices.

Findings

Disproved the strong conjecture for all graphs.

Confirmed the weak conjecture holds universally.

Established a majorization relationship between the indices.

Abstract

We resolve two conjectures of Hri\v{n}\'{a}kov\'{a}, Knor and \v{S}krekovski (2019) concerning the relationship between the variable Wiener index and variable Szeged index for a connected, non-complete graph, one of which would imply the other. The strong conjecture is that for any such graph there is a critical exponent in $(0,1]$, below which the variable Wiener index is larger and above which the variable Szeged index is larger. The weak conjecture is that the variable Szeged index is always larger for any exponent exceeding $1$. They proved the weak conjecture for bipartite graphs, and the strong conjecture for trees. In this note we disprove the strong conjecture, although we show that it is true for almost all graphs, and for bipartite and block graphs. We also show that the weak conjecture holds for all graphs by proving a majorization relationship.