Investigation of General Power Sum-Connectivity Index for Some Classes of Extremal Graphs

Abstract

In this work, we introduce a new topological index called a general power sum-connectivity index and we discuss this graph invariant for some classes of extremal graphs. This index is defined by $Y_{\alpha }\left(G\right)={\displaystyle {\sum }_{uv\in E\left(G\right)}{\left(d{\left(u\right)}^{d\left(u\right)}+d{\left(v\right)}^{d\left(v\right)}\right)}^{\alpha }}$ , where $d\left(u\right)$ and $d\left(v\right)$ represent the degree of vertices $u$ and $v$ , respectively, and $\alpha \ge 1$ . A connected graph $G$ is called a $k$ -generalized quasi-tree if there exists a subset $V_k\subset V\left(G\right)$ of cardinality $k$ such that the graph $G-V_k$ is a tree but for any subset $V_{k-1}\subset V\left(G\right)$ of cardinality $k-1$ , the graph $G-V_{k-1}$ is not a tree. In this work, we find a sharp lower and some sharp upper bounds for this new sum-connectivity index.