Zeroth-order general Randic index of $k$-generalized quasi trees

Abstract

For a simple graph $G(V,E)$, the zeroth-order general Randi\' c index is defined as ${}^0{R}_{\alpha }(G)={\sum }_{v\in V(G)}d(v{)}^{\alpha }$, where $d(v)$ is the degree of the vertex $v$ and $\alpha \ne 0$ is a real number. The $k$-generalized quasi-tree is a connected graph $G$ with a subset $V_k\subset V(G)$, where $|V_k|=k$ such that $G-V_k$ is a tree, but for any subset $V_{k-1}\subset V(G)$ with cardinality $k-1$, $G-V_{k-1}$ is not a tree. In this paper, we characterize the extremal $k$-generalized quasi trees with the minimum and maximum values of the zeroth-order general Randi\' c index for $\alpha \ne 0$.