Computational and analytical studies of the Randi\'c index in Erd\"os-R\'{e}nyi models

TL;DR

This paper investigates the Randić index in Erdős-Rényi random graphs, revealing how it scales with the average degree and identifying different graph regimes, along with deriving new relations and inequalities for this index.

Contribution

The study provides the first detailed scaling analysis of the Randić index in Erdős-Rényi models and introduces new relations and bounds for this topological index.

Findings

The Randić index scales with the product of the number of vertices and connection probability.

Three regimes of graph structure are identified based on the average degree: isolated, transition, and almost complete.

New inequalities relating the Randić index to other topological indices are derived.

Abstract

In this work we perform computational and analytical studies of the Randi\'c index $R(G)$ in Erd\"os-R\'{e}nyi models $G(n,p)$ characterized by $n$ vertices connected independently with probability $p \in (0,1)$. First, from a detailed scaling analysis, we show that $\left\langle \overline{R}(G) \right\rangle = \left\langle R(G)\right\rangle/(n/2)$ scales with the product $\xi\approx np$, so we can define three regimes: a regime of mostly isolated vertices when $\xi < 0.01$ ($R(G)\approx 0$), a transition regime for $0.01 < \xi < 10$ (where $0<R(G)< n/2$), and a regime of almost complete graphs for $\xi > 10$ ($R(G)\approx n/2$). Then, motivated by the scaling of $\left\langle \overline{R}(G) \right\rangle$, we analytically (i) obtain new relations connecting $R(G)$ with other topological indices and characterize graphs which are extremal with respect to the relations obtained and (ii) apply these results in order to obtain inequalities on $R(G)$ for graphs in Erd\"os-R\'{e}nyi models.