Sufficient conditions for Hamiltonianity in terms of the Zeroth-order General Randi\'c Index

For a (molecular) graph $G$ and any real number $\alpha\ne 0$ , the zero-order general Randić index , denote by $^0R_\alpha$, is defined by the following equation: \begin{align*}
{^0R_\alpha} (G) =\sum_{v\in G}d_G (v) ^{\alpha} (\alpha \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.