The variation of the Randic index with regard to minimum and maximum degree

TL;DR

This paper investigates the minimum variation of the Randić index in connected graphs with specified minimum and maximum degrees, identifying extremal graph structures and generalizing results for maximum degree constraints.

Contribution

It characterizes extremal graphs minimizing the Randić index variation based on minimum and maximum degree conditions, extending previous results.

Findings

Extremal graphs are complete split graphs for certain degree ranges.

The number of vertices of specific degrees in extremal graphs is explicitly determined.

Results are generalized for graphs with given maximum degree.

Abstract

The variation of the Randi\'c index $ R'(G) $ of a graph $G$ is defined by\ $R(G) = \sum_{uv \in E(G)}\frac 1{\max \{d(u) d(v)\}}$, where $d(u)$ is the degree of vertex $u$ and the summation extends over all edges $uv$ of $G$. Let $G(k,n)$ be the set of connected simple $n$-vertex graphs with minimum vertex degree $k$. In this paper we found in $G(k,n)$ graphs for which the variation of the Randi\'c index attains its minimum value. When $k \leq \frac n2$ the extremal graphs are complete split graphs $K_{k,n-k}^*$, which only vertices of two degrees, i.e. degree $k$ and degree $n-1$, and the number of vertices of degree $k$ is $n-k$, while the number of vertices of degree $n-1$ is $k$. For $k \geq \frac n2$ the extremal graphs have also vertices of two degrees $k$ and $n-1$, and the number of vertices of degree $k$ is $\frac n2$. Further, we generalized results for graphs with given maximum degree.