Comparing Wiener, Szeged and revised Szeged index on cactus graphs

TL;DR

This paper investigates relationships between the Wiener, Szeged, and revised Szeged indices on cactus graphs, establishing bounds, sharpness, and conditions for equality, with a new vertex-based formulation of the revised Szeged index.

Contribution

It provides new bounds and conditions for the indices on cactus graphs, including a novel vertex-based formulation of the revised Szeged index.

Findings

Szeged index is at most twice the Wiener index on cactus graphs.

Revised Szeged index is at least twice the Wiener index when all blocks are cycles.

Bounds are proven to be sharp with cases of equality examined.

Abstract

We show that on cactus graphs the Szeged index is bounded above by twice the Wiener index. For the revised Szeged index the situation is reversed if the graph class is further restricted. Namely, if all blocks of a cactus graph are cycles, then its revised Szeged index is bounded below by twice its Wiener index. Additionally, we show that these bounds are sharp and examine the cases of equality. Along the way, we provide a formulation of the revised Szeged index as a sum over vertices, which proves very helpful, and may be interesting in other contexts.