Sufficient conditions for Hamiltonianity in terms of the Zeroth-order General Randić Index

For a (molecular) graph $G$ and any real number $\alpha \mathrm{\ne }0$ , the zero-order general Randić index , denote by ${}^0{R}_{\alpha }$, is defined by the following equation: \begin{align*} {^0R_α} (G) =\sum_{v\in G}d_G (v) ^α (α\in $\mathbb{R}$-\left{0\right}) . \end{align*} In this paper, we use this index to give sufficient conditions for a graph $G$ to satisfy the Hamiltonian (or $k$-Hamiltonian) property, and show that none of these conditions can be dropped. Finally we give similar results for the case when $G$ is a balanced bipartite graph.