Block eccentricity, a radius bound, and an application to the Randi\'c index

TL;DR

This paper introduces a new framework for analyzing eccentricity through blocks, extending radius and center concepts, and applies it to derive bounds and verify conjectures related to the Randić index in graph theory.

Contribution

It extends eccentricity concepts to blocks, classifies graphs based on block and vertex radius relationships, and verifies a conjecture involving the Randić index.

Findings

Derived a new lower bound on diameter based on central block diameter.

Identified a subgraph that preserves vertex radius and respects block structure.

Verified a conjectured bound between vertex radius and the Randić index for cactus graphs.

Abstract

We propose a framework for thinking about eccentricity in terms of blocks. We extend the familiar definitions of radius and center to blocks and verify that a central block contains all central points. We classify graphs into two types depending upon the relationship between block radius and vertex radius and between central blocks and central vertices; from this we derive a new lower bound on diameter in terms of the diameter of the central block. We also identify a subgraph which respects the block structure of the original graph and realizes the same vertex radius, and we use it to verify that cactus graphs satisfy a conjectured bound between vertex radius and the Randic index, an invariant from mathematical chemistry.