On the zeroth-order general Randi\'c index, variable sum exdeg index and trees having vertices with prescribed degree

TL;DR

This paper characterizes trees with extremal zeroth-order general Randić index and variable sum exdeg index among trees with fixed parameters, providing new insights into their structural properties.

Contribution

It determines extremal trees for these indices within specific classes of trees with prescribed degrees and segments, extending previous knowledge on graph invariants.

Findings

Identifies trees with maximum and minimum indices in various classes.

Shows extremal trees for one class are also extremal for another.

Provides formulas relating segments and degree-2 vertices in trees.

Abstract

The zeroth-order general Randi\'c index (usually denoted by $R_{\alpha}^{0}$) and variable sum exdeg index (denoted by $SEI_{a}$) of a graph $G$ are defined as $R_{\alpha}^{0}(G)= \sum_{v\in V(G)} (d_{v})^{\alpha}$ and $SEI_{a}(G)= \sum_{v\in V(G)}d_{v}a^{d_{v}}$ where $d_{v}$ is degree of the vertex $v\in V(G)$, $a$ is a positive real number different from 1 and $\alpha$ is a real number other than $0$ and $1$. A segment of a tree is a path $P$, whose terminal vertices are branching or pendent, and all non-terminal vertices (if exist) of $P$ have degree 2. For $n\ge6$, let $\mathbb{PT}_{n,n_1}$, $\mathbb{ST}_{n,k}$, $\mathbb{BT}_{n,b}$ be the collections of all $n$-vertex trees having $n_1$ pendent vertices, $k$ segments, $b$ branching vertices, respectively. In this paper, all the trees with extremum (maximum and minimum) zeroth-order general Randi\'c index and variable sum exdeg index are determined from the collections $\mathbb{PT}_{n,n_1}$, $\mathbb{ST}_{n,k}$, $\mathbb{BT}_{n,b}$. The obtained extremal trees for the collection $\mathbb{ST}_{n,k}$ are also extremal trees for the collection of all $n$-vertex trees having fixed number of vertices with degree 2 (because it is already known that the number of segments of a tree $T$ can be determined from the number of vertices of $T$ with degree 2 and vise versa).