Ratio of the number of 1-nearly independent vertex subsets and the Merrifield-Simmons index
Pith reviewed 2026-05-10 17:50 UTC · model grok-4.3
The pith
The ratio of 1-nearly independent vertex subsets to the Merrifield-Simmons index has sharp bounds for connected graphs, trees and forests
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We establish sharp lower and upper bounds for the ratio σ₁(G)/σ₀(G) over the classes of connected graphs, trees, and forests.
What carries the argument
The ratio σ₁(G)/σ₀(G) that compares the count of 1-nearly independent subsets to the count of independent subsets.
Load-bearing premise
The assumption that the minimal and maximal ratios are attained by some graphs within the given classes and that these can be determined exactly.
What would settle it
To falsify the claim, one would need to exhibit a connected graph, tree, or forest for which the computed ratio lies outside the lower or upper bound given in the paper.
read the original abstract
The number $\sigma_k(G)$ of induced subgraphs with size $k$ of a graph $G$ was introduced recently as the number of $k$-nearly independent vertex subsets of $G$. Results highlighting similarity and difference in the behaviours of $\sigma_1$ and $\sigma_0$, have been reported. In this paper, we provide more comparison tools, by studying the ratio $\frac{\sigma_1(G)}{\sigma_{0}(G)}$. We establish sharp lower and upper bounds for this ratio over various classes of graphs, including connected graphs, trees, and forests.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper defines σ₀(G) as the Merrifield-Simmons index (number of independent sets) and σ₁(G) as the number of 1-nearly independent vertex subsets (induced subgraphs with at most one edge). It establishes sharp lower and upper bounds on the ratio σ₁(G)/σ₀(G) for the classes of connected graphs, trees, and forests, with equality cases attained at stars, paths, and disjoint unions of paths. The proofs proceed by structural induction on leaves and pendant edges together with exhaustive case analysis on neighborhoods.
Significance. The sharp bounds supply concrete comparison tools between σ₁ and σ₀ on standard graph families, extending earlier observations on similarities and differences in their behavior. The explicit extremal graphs and closed-form recursions that close without residual cases constitute a solid contribution to extremal enumeration in graph theory.
minor comments (3)
- [§2] §2: The definition of σ₁(G) would be clearer if accompanied by a small concrete example (e.g., for P₃ or K_{1,3}) showing which subsets are counted.
- [Theorem 3.1] Theorem 3.1: The equality-attaining graphs (stars) are stated only in the proof; moving the equality cases into the theorem statement itself would improve readability.
- [Theorems 4.2 and 5.3] The induction hypotheses in the proofs of Theorems 4.2 and 5.3 are stated clearly, but a brief remark on why the base cases for n=1,2 are exhaustive would eliminate any possible reader uncertainty.
Simulated Author's Rebuttal
We thank the referee for the positive evaluation of our manuscript and for recommending minor revision. The referee's summary correctly captures the definitions, the sharp bounds obtained for connected graphs, trees, and forests, and the proof techniques employed.
Circularity Check
No significant circularity detected
full rationale
The paper defines σ₀(G) as the Merrifield-Simmons index (standard independent-set count) and σ₁(G) as the count of vertex subsets inducing at most one edge. It then derives sharp bounds on the ratio via structural induction, leaf deletion, and exhaustive neighborhood case analysis for connected graphs, trees, and forests, with equality cases at stars, paths, and path forests. No parameter is fitted to data and then renamed a prediction; no uniqueness theorem is imported from the authors' prior work; no ansatz is smuggled via self-citation; and the extremal characterizations follow directly from the recursive relations without reducing to the input definitions by construction. The derivation is therefore self-contained against the combinatorial definitions.
Axiom & Free-Parameter Ledger
Lean theorems connected to this paper
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IndisputableMonolith/Foundation/RealityFromDistinction.leanreality_from_one_distinction unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
Theorem 2: Q(G) ≥ (n−1)/(2^{n−1}+1) for connected G, equality iff G = S_n
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- extends
- The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
- uses
- The paper appears to rely on the theorem as machinery.
- contradicts
- The paper's claim conflicts with a theorem or certificate in the canon.
- unclear
- Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.
Reference graph
Works this paper leans on
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[1]
E. O. D. Andriantiana, and Z. B. Shozi. The number of 1-nearly independent vertex subsets.Quaes- tiones Mathematicae47(12), 2353–2373, 2024
work page 2024
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[4]
L. Abrams, and L.-K. Lauderdale. On a ratio of Wiener indices for embedded graphs.Discrete Math- ematics, 346(5):113320, 2023
work page 2023
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[5]
R. E. Merrifield, and H. E. Simmons. Enumeration of structure-sensitive graphical subsets: Theory. Proceedings of the National Academy of Sciences78(2), 692–695, 1981
work page 1981
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[6]
S. Wagner. Correlation of graph-theoretical indices.SIAM Journal on Discrete Mathematics21(1):33– 46, 2007
work page 2007
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[7]
S. Wagner, and I. Gutman. Maxima and minima of the Hosoya index and the Merrifield-Simmons index: a survey of results and techniques.Acta Applicandae Mathematicae112(3), 323–346, 2010. Audace A. V. Dossou-Olory, Institut National de l’Eau, Abomey-Calavi, and Institut de Math´ematiques et de Sciences Physiques, Dangbo, Universit ´e d’Abomey-Calavi, B´enin ...
work page 2010
discussion (0)
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