Some Bond Incident Degree Indices of (Molecular) Graphs with Fixed Order and Number of Cut Vertices

Abstract

A bond incident degree (BID) index of a graph $G$ is defined as ${\displaystyle {\sum }_{uv\in E\left(G\right)}f\left(d_G\left(u\right),d_G\left(v\right)\right)}$ , where $d_G\left(w\right)$ denotes the degree of a vertex $w$ of $G$ , $E\left(G\right)$ is the edge set of $G$ , and $f$ is a real-valued symmetric function. The choice $f\left(d_G\left(u\right),d_G\left(v\right)\right)=a^{d_G\left(u\right)}+a^{d_G\left(v\right)}$ in the aforementioned formula gives the variable sum exdeg index ${\text{SEI}}_a$ , where $a\ne 1$ is any positive real number. A cut vertex of a graph $G$ is a vertex whose removal results in a graph with more components than $G$ has. A graph of maximum degree at most 4 is known as a molecular graph. Denote by ${\mathbb{V}}_{n,k}$ the class of all $n$ -vertex graphs with $k\ge 1$ cut vertices and containing at least one cycle. Recently, Du and Sun [AIMS Mathematics, vol. 6, pp. 607–622, 2021] characterized the graphs having the maximum value of ${\text{SEI}}_a$ from the set ${\mathbb{V}}_n^k$ for $a>1$ . In the present paper, we not only characterize the graphs with the minimum value of ${\text{SEI}}_a$ from the set ${\mathbb{V}}_n^k$ for $a>1$ , but we also solve a more general problem concerning a special type of BID indices. As the obtained extremal graphs are molecular graphs, they remain extremal if one considers the class of all $n$ -vertex molecular graphs with $k\ge 1$ cut vertices and containing at least one cycle.