Some Vertex/Edge-Degree-Based Topological Indices of r -Apex Trees

Abstract

In chemical graph theory, graph invariants are usually referred to as topological indices. For a graph $G$ , its vertex-degree-based topological indices of the form $\text{BID}\left(G\right)={\displaystyle {\sum }_{uv\in E\left(G\right)}\beta \left(d_u,d_v\right)}$ are known as bond incident degree indices, where $E\left(G\right)$ is the edge set of $G$ , $d_w$ denotes degree of an arbitrary vertex $w$ of $G$ , and $\beta$ is a real-valued-symmetric function. Those $\text{BID}$ indices for which $\beta$ can be rewritten as a function of $d_u+d_v-2$ (that is degree of the edge $uv$ ) are known as edge-degree-based $\text{BID}$ indices. A connected graph $G$ is said to be $r$ -apex tree if $r$ is the smallest nonnegative integer for which there is a subset $R$ of $V\left(G\right)$ such that $\left|R\right|=r$ and $G-R$ is a tree. In this paper, we address the problem of determining graphs attaining the maximum or minimum value of an arbitrary $\text{BID}$ index from the class of all $r$ -apex trees of order $n$ , where $r$ and $n$ are fixed integers satisfying the inequalities $n-r\ge 2$ and $r\ge 1$ .