Randic index, radius, and diameter for cactus graphs

TL;DR

This paper investigates bounds on the Randic index for cactus graphs, confirming a radius-related conjecture and strengthening diameter bounds, while also deriving new bounds based on graph properties.

Contribution

It verifies the radius bound conjecture for cacti and improves diameter bounds, also providing new bounds for the Randic index in specific cactus subclasses.

Findings

Confirmed the radius bound for cacti.

Strengthened diameter bounds for cactus graphs.

Derived new bounds for the Randic index based on graph properties.

Abstract

We study the Randic index for cactus graphs. It is conjectured to be bounded below by radius (for other than an even path), and it is known to obey several bounds based on diameter. We study radius and diameter for cacti then verify the radius bound and strengthen two diameter bounds for cacti. Along the way, we produce several other bounds for the Randic index in terms of graph size, order, and valency for several special classes of graphs, including chemical nontrivial cacti and cacti with starlike BC-trees.