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

Abstract: For a (molecular) graph $G$ and any real number $α\ne 0$ , the zero-order general Randić index , denote by $^0R_α$, 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.

• The paper establishes sharp, degree-based sufficient conditions for k-Hamiltonicity using the zero-order general Randić index.
• It employs convex analysis and extremal constructions to derive unified thresholds that generalize classical indices like Zagreb and Forgotten.
• The study provides explicit asymptotic bounds and extensions to balanced bipartite graphs, offering a versatile framework for graph theory and chemical applications.

Sufficient Conditions for Hamiltonianity in Terms of the Zeroth-order General Randić Index

Introduction and Context

The paper "Sufficient conditions for Hamiltonianity in terms of the Zeroth-order General Randić Index" (2604.02254) presents a comprehensive investigation into the relationship between Hamiltonian properties of graphs and the zero-order general Randić index, denoted as $^0R_\alpha(G)$, for arbitrary real $\alpha \ne 0$. The study builds on prior work connecting degree-based topological indices and Hamiltonicity, encompassing earlier sufficient conditions stated in terms of indices such as the Wiener, Harary, Zagreb, and Forgotten indices. This work generalizes these results, offering unified, parameterized sufficient conditions for (k-)Hamiltonicity for general simple graphs and balanced bipartite graphs.

The $^0R_\alpha$ index, a degree-based graph invariant, is defined as:

$^0R_\alpha(G) = \sum_{v \in V(G)} d_G(v)^{\alpha}$

where $d_G(v)$ is the degree of $v$ in $G$, and $\alpha \in \mathbb{R} \setminus \{0\}$. This index is relevant not only for mathematical chemistry but also as a unifying tool encompassing various classical indices, e.g., for $\alpha=1$ it is twice the edge count, for $\alpha=2$ it is the first Zagreb index, and for $\alpha \ne 0$0 it is the Forgotten index.

Main Results: Sufficient Conditions via $\alpha \ne 0$1

General Graphs

The central contribution is the derivation of sharp sufficient conditions for a simple connected graph $\alpha \ne 0$2 of order $\alpha \ne 0$3 to be $\alpha \ne 0$4-Hamiltonian (in particular Hamiltonian for $\alpha \ne 0$5), parameterized by the real exponent $\alpha \ne 0$6 (and, in a dual sense, for $\alpha \ne 0$7). These conditions take the following general form:

• For $\alpha \ne 0$8, if

$\alpha \ne 0$9

then $^0R_\alpha$0 is $^0R_\alpha$1-Hamiltonian.

Here, $^0R_\alpha$2 is an explicitly derived function of $^0R_\alpha$3 encoding extremal degree sequences that approach the Chvátal-type degree threshold for $^0R_\alpha$4-Hamiltonicity. Importantly, the authors prove the sharpness of these bounds, exhibiting that they cannot be lowered without admitting non-Hamiltonian graphs.

These results provide a form of Hamiltonicity threshold governed by the summability of vertex degrees raised to a real power, capturing a spectrum of existing indices as special cases. For $^0R_\alpha$5, the results yield conditions for Hamiltonicity.

For negative $^0R_\alpha$6, analogous conditions with reversed inequalities are shown.

Sharpness and Extremal Structures

The paper demonstrates that the extremal examples achieving equality in these bounds have unique graph structures (such as $^0R_\alpha$7), strengthening the assertion that the provided sufficient conditions are exact and not merely bounding heuristics.

The Case $^0R_\alpha$8 and Asymptotics

For large enough $^0R_\alpha$9 (in relation to $^0R_\alpha(G) = \sum_{v \in V(G)} d_G(v)^{\alpha}$0), the results are refined: for $^0R_\alpha(G) = \sum_{v \in V(G)} d_G(v)^{\alpha}$1 and sufficiently large $^0R_\alpha(G) = \sum_{v \in V(G)} d_G(v)^{\alpha}$2, the theorem establishes that a simple, explicit degree-sum bound in terms of $^0R_\alpha(G) = \sum_{v \in V(G)} d_G(v)^{\alpha}$3 suffices. Through detailed asymptotic analysis, threshold functions for $^0R_\alpha(G) = \sum_{v \in V(G)} d_G(v)^{\alpha}$4 relative to $^0R_\alpha(G) = \sum_{v \in V(G)} d_G(v)^{\alpha}$5 are derived:

$^0R_\alpha(G) = \sum_{v \in V(G)} d_G(v)^{\alpha}$6

serves as a threshold between the two types of sufficient conditions (centered at minimum and maximum degree extremal sequences). The authors demonstrate that outside an $^0R_\alpha(G) = \sum_{v \in V(G)} d_G(v)^{\alpha}$7 interval for $^0R_\alpha(G) = \sum_{v \in V(G)} d_G(v)^{\alpha}$8, one of the two $^0R_\alpha(G) = \sum_{v \in V(G)} d_G(v)^{\alpha}$9 (or $d_G(v)$0 for Hamiltonicity) terms dominates.

Hamiltonicity in Balanced Bipartite Graphs

Specialization to balanced bipartite graphs yields parallel results. If $d_G(v)$1 is a connected balanced bipartite graph of order $d_G(v)$2, $d_G(v)$3 exceeding (or, for $d_G(v)$4, falling below) a specific threshold implies Hamiltonicity. The extremal construction for every case is sharp, with the combinatorial structure corresponding to bipartite graphs with specific edge deletions.

Unified View with Previous Degree-based Indices

Importantly, the framework of the $d_G(v)$5 index encompasses previously proposed sufficient conditions:

• $d_G(v)$6 recovers edge-count-based conditions,
• $d_G(v)$7 links directly to Zagreb index results,
• $d_G(v)$8 recovers conditions via the Forgotten index,
• Variable $d_G(v)$9 interpolates between or strengthens these classical cases.

Technical Approach

The derivation is based on reducing the Hamiltonicity (or $v$0-Hamiltonicity) problem to degree sequence conditions of Chvátal type (i.e., bounding the degree sequences using generalized positional arguments). The authors compute the maximal value of the $v$1 index on graphs forbidden from being $v$2-Hamiltonian, obtaining tight bounds by convexity arguments and explicit maximization over possible extremal degree sequences (via calculus and combinatorial analysis). Lemmas show that the bounding functions are convex in the region of interest, warranting that the maxima or minima are obtained at endpoints. The arguments utilize a blend of asymptotic analysis, inequalities (including Jensen and Taylor expansion), and combinatorial constructions.

For the bipartite case, Chvátal’s corresponding result is invoked, and similar degree sequence analysis leads to tight bounds expressed via another family of parameterizations (denoted $v$3).

Implications and Directions for Future Research

These results significantly broaden the landscape of analytical connections between degree-based graph invariants and Hamiltonian properties, strengthening the theoretical toolkit for attacking both extremal graph-theoretic and molecular chemistry problems. The parameterization via $v$4 allows for tailoring sufficient conditions to the context (e.g., physical, chemical, or combinatorial) at hand.

The paper prompts several potential research directions:

• Determination of the best possible threshold functions for $v$5 as a function of $v$6 beyond the leading order provided.
• Extension of these sufficient conditions to randomized graph models where degree sequences are governed by probability distributions.
• Investigation into analogous necessary conditions, possibly via stability analysis near the extremal examples.
• Application to other important graph structures (e.g., $v$7-connected, traceable graphs) and to directed graphs via out-/in-degree versions of $v$8.

Conclusion

This work offers an explicit, sharp family of sufficient conditions for assertion of Hamiltonicity and $v$9-Hamiltonicity in general and balanced bipartite graphs via the zero-order general Randić index $G$0. The generality and precision of the results, coupled with careful extremal and asymptotic analysis, provide a unifying and extendible framework subsuming and refining earlier degree-based conditions. The analytical characterization of the threshold effect—depending on both graph order and the exponent $G$1—adds notable theoretical insight. The results are immediately relevant for both extremal graph theory and applied chemical graph theory analyses.