On the zeroth-order general Randić index, variable sum exdeg index and trees having vertices with prescribed degree

Abstract

The zeroth-order general Randić index (usually denoted by $R_{\alpha }^0$) and variable sum exdeg index (denoted by $SE{I}_a$) of a graph $G$ are defined as $R_{\alpha }^0(G)={\sum }_{v\in V(G)}(d_v{)}^{\alpha }$ and $SE{I}_a(G)={\sum }_{v\in V(G)}{d}_v{a}^{d_v}$ 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 $n\ge 6$, 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 Randić 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).

10 pages

References in corpus (1)

• On the Extremal Graphs with Respect to Bond Incident Degree Indices