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

TL;DR

This paper establishes sufficient conditions based on the zero-order general Randić index for a graph to be Hamiltonian, including special cases for balanced bipartite graphs, and demonstrates that these conditions are necessary.

Contribution

The paper introduces new Hamiltonianity conditions derived from the zero-order general Randić index and proves their necessity, extending to balanced bipartite graphs.

Findings

Derived sufficient conditions for Hamiltonian graphs using the Randić index.

Proved that these conditions are necessary and cannot be omitted.

Extended results to balanced bipartite graphs.

Abstract

For a (molecular) graph $G$ and any real number $\alpha\ne 0$ , the zero-order general Randi\'c 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.