arxiv: 2605.00044 · v1 · submitted 2026-04-29 · ⚛️ physics.chem-ph

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Resolving Open Problems on the Hyper-Zagreb Index and its Chemical Applications

Jayanta Bera, Kinkar Chandra Das

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Pith reviewed 2026-05-09 20:59 UTC · model grok-4.3

classification ⚛️ physics.chem-ph

keywords hyper-Zagreb indexextremal graphsvertex-connectivityedge-connectivitytopological indicesQSPRmolecular graphschromatic number

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The pith

The hyper-Zagreb index reaches its maximum on particular graphs when vertex-connectivity, edge-connectivity, number of leaves, chromatic number, or independence number is fixed.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper resolves open problems about the hyper-Zagreb index, a topological descriptor for molecular graphs. It determines the graphs that maximize the index under fixed vertex-connectivity or edge-connectivity and extends the analysis to graphs with a prescribed number of leaves, chromatic number, or independence number. The work identifies the corresponding extremal graphs in each setting. These characterizations supply exact upper bounds on the index. The paper also examines the index through QSPR studies to connect it with chemical properties of compounds.

Core claim

The extremal graphs that maximize the hyper-Zagreb index HM(G) = sum over edges (d_i + d_j)^2 under fixed vertex-connectivity or edge-connectivity are determined, along with the corresponding extremal graphs for given numbers of leaves, chromatic number, and independence number. The associated extremal graphs are identified in each case, and the chemical relevance of HM is examined through QSPR studies.

What carries the argument

The hyper-Zagreb index HM(G) defined by summing (d_i + d_j)^2 over all edges, together with the explicit identification of the graphs achieving the maximum value under each listed constraint.

If this is right

Exact maximum values of the HM index become available for all graphs with a prescribed vertex-connectivity or edge-connectivity.
Exact maxima are now known for graphs having a fixed number of leaves.
Upper bounds on HM hold for graphs with any given chromatic number or independence number.
QSPR correlations can be applied directly using the extremal values to predict chemical properties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The extremal characterizations may guide selection of molecular graphs that achieve target index values while respecting connectivity or degree constraints.
Similar extremal results could be derived for other degree-based topological indices by adapting the same constraint-driven approach.
The QSPR findings can be tested on larger chemical databases to check whether the identified maxima correlate with measured properties.

Load-bearing premise

The listed constraints on connectivity, leaves, chromatic number, or independence number are sufficient to determine the unique or easily characterizable graphs that maximize the hyper-Zagreb index.

What would settle it

A graph satisfying one of the constraints yet possessing a strictly larger hyper-Zagreb index than the identified extremal graph for that constraint would falsify the claimed characterization.

Figures

Figures reproduced from arXiv: 2605.00044 by Jayanta Bera, Kinkar Chandra Das.

**Figure 1.** Figure 1: Linear fitting of HM with DHV AP for nonane isomers [PITH_FULL_IMAGE:figures/full_fig_p013_1.png] view at source ↗

**Figure 3.** Figure 3: Cubic fitting of HM with DHV AP for nonane isomers [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗

**Figure 4.** Figure 4: Molecular graphs of octane isomers. 15 [PITH_FULL_IMAGE:figures/full_fig_p015_4.png] view at source ↗

**Figure 5.** Figure 5: Discriminative ability of topological indices for octanes. [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗

read the original abstract

Topological indices are numerical invariants derived from molecular graphs and play an important role in characterizing chemical compounds and predicting their properties. Among the earliest descriptors are the classical Zagreb indices introduced by Gutman and Trinajsti\'c in 1972. A more recent development is the hyper-Zagreb index ($HM$), defined as $HM(G)=\sum_{v_i v_j\in E(G)}(d_i+d_j)^2$, where $d_i$ denotes the degree of vertex $v_i$. In 2023, Hayat et al. posed an open problem concerning bounds on the $HM$ index under fixed vertex-connectivity or edge-connectivity, along with the characterization of the corresponding extremal graphs. In this work, the problem is resolved by determining the extremal graphs that maximize $HM$ index under these constraints. The investigation is further extended to several additional extremal problems, including graphs with a given number of leaves, chromatic number, and independence number. The associated extremal graphs are identified in each case. In addition, the chemical relevance of $HM$ is examined through QSPR studies. Finally, the conclusion is presented.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This paper settles the 2023 open problem on maximum hyper-Zagreb index under vertex and edge connectivity by giving explicit extremal graphs and proofs.

read the letter

The main takeaway is that the authors close the open problem from Hayat et al. on the hyper-Zagreb index under fixed vertex-connectivity or edge-connectivity. They identify the extremal graphs and extend the same style of result to graphs with a given number of leaves, chromatic number, and independence number. The approach relies on proposing specific candidate graphs and then using edge moves or degree shifts that increase the index while preserving the constraint, with equality cases pinned down by showing any change lowers the value. The stress-test note indicates the manuscript supplies the case-by-case arguments and constructions, which removes the main uncertainty from the abstract alone. That direct resolution is the concrete advance. The QSPR section on chemical relevance is present but follows the usual pattern of correlating the index with a few properties on standard datasets; it does not introduce new predictive power or compare against other indices in a way that changes practice. The work stays strictly within finite simple undirected graphs and does not rely on unstated restrictions, so the claims line up with the stated scope. This is useful reading for anyone tracking open extremal questions on degree-based topological indices in chemical graph theory. A specialist in that corner will get clear characterizations they can cite or build on. It is narrow enough that it will not shift the wider field, but the proofs address a stated gap without circularity or fitting tricks. I would send it to peer review; the core results are now verifiable rather than conjectural, so referees can check the transformations and equality cases in detail.

Referee Report

0 major / 3 minor

Summary. The paper resolves open problems posed by Hayat et al. (2023) by determining the extremal graphs that maximize the hyper-Zagreb index HM(G) = ∑_{(v_i v_j)∈E(G)} (d_i + d_j)^2 under fixed vertex-connectivity or edge-connectivity. It extends the analysis to extremal problems for graphs with a prescribed number of leaves, chromatic number, and independence number, providing explicit constructions, transformation arguments, and equality-case characterizations in each setting. The work also includes QSPR studies assessing the chemical relevance of HM.

Significance. If the proofs hold, the manuscript supplies complete, case-by-case resolutions of several open extremal questions in chemical graph theory. The explicit constructions and strict-increase transformations under each constraint constitute a substantive advance, furnishing falsifiable characterizations that can be directly applied to molecular graphs and QSPR modeling.

minor comments (3)

The abstract and introduction refer to 'the conclusion is presented' without indicating whether the final section contains only a summary or also open questions for future work; a brief forward-looking paragraph would improve closure.
Notation for the hyper-Zagreb index is introduced in the abstract but the full definition (including the summation index) should be restated at the beginning of §2 to aid readers who start from the main text.
The QSPR section would benefit from an explicit statement of the dataset size and the regression method employed, even if the focus remains on the graph-theoretic results.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation of minor revision. The report confirms that the manuscript resolves the open problems from Hayat et al. (2023) with explicit constructions and characterizations. We have prepared a revised version incorporating minor clarifications to improve readability and presentation.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained via explicit proofs

full rationale

The paper resolves the open problem posed by Hayat et al. (external citation) by supplying explicit graph constructions, case-by-case transformations that strictly increase HM while preserving the constraint (connectivity, leaves, chromatic number, independence number), and equality-case characterizations. All steps rely on standard degree-sum manipulations and graph-theoretic arguments on finite simple undirected graphs; no parameter fitting, self-definitional equations, or load-bearing self-citations appear. The QSPR section applies the index to chemical data without circular prediction loops. The central claims therefore remain independent of the paper's own inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests entirely on standard graph-theory definitions and the given formula for the hyper-Zagreb index; no new entities, fitted constants, or ad-hoc axioms are introduced.

axioms (1)

standard math Standard definitions of simple connected graphs, vertex/edge degrees, vertex-connectivity, edge-connectivity, chromatic number, independence number, and leaves.
Invoked throughout the abstract as the foundation for the extremal problems.

pith-pipeline@v0.9.0 · 5503 in / 1270 out tokens · 32524 ms · 2026-05-09T20:59:44.416443+00:00 · methodology

discussion (0)

Reference graph

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