On the Randi\'{c} index and conditional parameters of a graph

TL;DR

This paper investigates graph parameters related to vertex degrees, focusing on the Randić index and other conditional measures, using matrix analysis to derive bounds and explore their properties.

Contribution

It introduces a matrix-based approach to analyze the Randić index and related parameters, providing new bounds and insights into their behavior.

Findings

Derived tight bounds on the Randić index and related parameters.

Connected eigenvalues of matrix ${\

to the studied graph parameters.

Abstract

The aim of this paper is to study some parameters of simple graphs related with the degree of the vertices. So, our main tool is the $n\times n$ matrix ${\cal A}$ whose ($i,j$)-entry is $$ a_{ij}= \left\lbrace \begin{array}{ll} \frac{1}{\sqrt{\delta_i\delta_j}} & {\rm if }\quad v_i\sim v_j ; \\ 0 & {\rm otherwise,} \end{array} \right. $$ where $\delta_i$ denotes the degree of the vertex $v_i$. We study the Randi\'{c} index and some interesting particular cases of conditional excess, conditional Wiener index, and conditional diameter. In particular, using the matrix ${\cal A}$ or its eigenvalues, we obtain tight bounds on the studied parameters.