Extremal Zagreb indices of bicyclic hypergraphs

Changjiang Bu; Hong Zhou

REVIEW 3 major objections 4 minor 1 cited by

Extremal Zagreb indices of bicyclic hypergraphs

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read For $m\ge 6$, the hypergraph $C^k_{2,n,1,2,1}(m-4)$ attains the maximum Zagreb index among all linear bicyclic $k$-uniform hypergraphs.

desk verdict The extremal problem is natural and the answers are plausible, but the proof leans on an unproved structural classification and a hand-wavy compression argument. read the letter →

arxiv 2506.08875 v1 pith:QRS2DNL2 submitted 2025-06-10 math.CO

Hong Zhou , Changjiang Bu This is my paper

classification math.CO MSC 05C6505C09

keywords Zagrebindexbicyclichypergraphlineark-uniformextremaledgemovingoperationgirth

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper asks which linear $k$-uniform bicyclic hypergraph maximizes or minimizes the Zagreb index, the sum of the squares of vertex degrees. It claims that for $m\ge 6$ edges, the maximum is always attained by one explicit hypergraph, denoted $C^k_{2,n,1,2,1}(m-4)$, built from short hypercycles joined through a common vertex and carrying all surplus edges as pendant edges. The minimum is achieved by any hypergraph in the class whose maximum degree is $2$, with value $3km-2n$. The paper also settles the girth-constrained version: for even girth $g$ the extremal hypergraph is $C^k_{1,n,g/2,g/2,g/2}(m-3g/2)$, and for odd girth $g$ it is $C^k_{2,n,\lfloor g/2\rfloor,\lceil g/2\rceil,\lfloor g/2\rfloor}(m-g-\lfloor g/2\rfloor)$. If correct, this closes the extremal question for the whole class of linear bicyclic uniform hypergraphs for all but finitely many edge counts.

What carries the argument

The load-bearing object is Lemma 2.1: moving $t$ edges incident to a vertex $u$ to a lower-degree vertex $v$ changes the Zagreb index by $2t(t + d_H(v)-d_H(u))$, which is strictly positive when $d_H(v)>d_H(u)-t$. The paper uses this operation to shepherd any hypergraph in the structural families $B_n^k$ and $C_n^k$ toward a canonical shape while monotonically increasing $M(H)$; the remaining comparison is a calculus problem on single-variable polynomials in the girth $g$ and edge count $m$. The classification of linear bicyclic $k$-uniform hypergraphs into $B_n^k$ and $C_n^k$ is the stage on which the operation runs.

What would settle it

Run an exhaustive computer search for $k=3$, $m=6$, $n=11$, covering all linear $3$-uniform bicyclic hypergraphs with $6$ edges and $11$ vertices, and check whether any hypergraph has Zagreb index larger than $C^k_{2,11,1,2,1}(2)$ or fails to belong to $B_n^k\cup C_n^k$; either finding would refute Theorem 3.5 or the Section 2 classification.

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Extended reading notes

Core claim

The paper establishes that among all connected linear $k$-uniform hypergraphs with $n$ vertices and $m$ edges satisfying $n=m(k-1)-1$, the maximum of $M(H)=\sum_{u} d_H(u)^2$ is achieved by $C^k_{2,n,1,2,1}(m-4)$ whenever $m\ge 6$. This hypergraph is the explicit member of the family $C^k_n$ with parameters $p=1$, $q=2$, $l=1$, with $m-4$ pendant edges attached at a degree-$3$ vertex. The minimum is less sharp: any such hypergraph with maximum degree $2$ attains the minimum, whose value is $3km-2n$. The proof reduces every admissible hypergraph, by a strictly increasing edge-moving operation, to one of finitely many comparison shapes, then compares the resulting quadratic expressions in $m$ and the girth $g$.

Load-bearing premise

The argument leans on the Section 2 assertion, stated without proof or citation, that every connected linear $k$-uniform hypergraph with $n=m(k-1)-1$ belongs to one of the listed families $B_n^k$ or $C_n^k$; if that classification misses a bicyclic hypergraph, the maximum theorems do not cover it.

Editorial extensions

If this is right

For every $m\ge 6$, the maximum Zagreb index over the entire class is known explicitly: the winner is always $C^k_{2,n,1,2,1}(m-4)$.
For fixed girth $g$ and $m\ge 2g$, the extremal hypergraphs are $C^k_{1,n,g/2,g/2,g/2}(m-3g/2)$ for even $g$ and $C^k_{2,n,\lfloor g/2\rfloor,\lceil g/2\rceil,\lfloor g/2\rfloor}(m-g-\lfloor g/2\rfloor)$ for odd $g$.
Every linear bicyclic $k$-uniform hypergraph with maximum degree $2$ has the same minimum Zagreb index $3km-2n$, independent of its detailed shape.
Any hypergraph with a vertex of degree at least $3$ has strictly larger Zagreb index than every degree-$2$ hypergraph with the same parameters.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The paper does not pursue it, but the edge-moving lemma applies to any linear $k$-uniform hypergraph, so the same monotone-rearrangement strategy should yield extremal shapes for $r$-cyclic hypergraphs whenever a structural classification is available; the hard part would be the classification, not the inequality.
A direct computational check for small parameters, such as all linear $3$-uniform hypergraphs with $m=6$ and $n=11$, would test both the Section 2 classification and the extremal winner; the paper contains no such enumeration.
The minimum result suggests a purely degree-sequence characterization of minimal Zagreb index for sparse uniform hypergraphs, since the cycle structure drops out entirely when the maximum degree is $2$.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The result is a natural extension of known extremal results for unicyclic hypergraphs, and the claimed extremal shapes are plausible. The algebraic comparisons are careful and the statements are clean. But the proof as written rests on two substantial gaps: the structural classification of all linear bicyclic k-uniform hypergraphs into the B and C families is asserted without proof, and the "repeating the operation of Lemma 2.1" steps are not made rigorous.

On the first: Section 2 says all such hypergraphs consist of types B_n^k and C_n^k, but gives no argument or citation. The main theorems only establish the maximum within these families, so if the classification is incomplete the main result is unproven. This is load-bearing.

On the second: Lemma 2.1 shows a single edge-move raises the Zagreb index, but it does not guarantee the moved hypergraph stays linear. It can fail to: moving one edge of a 3-uniform hypercycle from a degree-2 vertex to the other degree-2 vertex can make two edges intersect in two vertices. The stress-test example is valid. That alone is not fatal—the proof only needs the final hypergraph to be in the class, not each intermediate—but the paper never actually proves that the sequence of moves exists and terminates at the claimed extremal hypergraph. Saying "obviously" does not cover the cases where the move changes the girth or creates an edge intersection that blocks the next planned move. This is the real soft spot.

There are also smaller issues: Lemma 2.1's wording "the edges incident with u" is ambiguous (should be "some edges"), and the paper never tracks connectedness through the moves. These are fixable.

What the paper does well: the M computations in Theorems 3.3–3.5 are explicit and check out on sample values; the comparison between B and C families is handled with actual inequalities. The problem is worth solving and the authors seem to know the target shapes. But the proof is not complete as written.

I'd send this to review, because the result is likely correct and the gaps are repairable. A serious referee should ask for a proof or citation of the classification and a more careful description of the compression sequences, including why the final hypergraph is in the right family. With that, the paper would be a solid contribution to a specialized area.

Referee Report

3 major / 4 minor

Summary. The paper studies the Zagreb index (sum of squared vertex degrees) of linear k-uniform bicyclic hypergraphs, i.e., connected k-uniform hypergraphs with n vertices and m edges satisfying n = m(k-1) - 1. It states that all such hypergraphs belong to two families, B_n^k and C_n^k, and then claims to identify, within these families, the hypergraphs attaining the minimum and maximum Zagreb index. The main results are Theorem 3.1 (minimum is attained by any hypergraph with maximum degree 2), Theorem 3.3 (maximum within B_n^k is B^k_{1,n,3,0,3}(m-6) for m >= 6), Theorem 3.4 (maximum within C_n^k is C^k_{2,n,1,2,1}(m-4) for m >= 6), and Theorem 3.5 (the same hypergraphs maximize globally). The proofs rely on the unproved B/C classification and on repeated applications of an edge-moving operation (Lemma 2.1) that strictly increases the Zagreb index.

Significance. If the results are correct, the paper would resolve the extremal problem for the Zagreb index among linear bicyclic uniform hypergraphs, extending known results for hypertrees and linear unicyclic hypergraphs. The explicit extremal structures and closed-form Zagreb indices are useful. The paper also correctly identifies that the minimum depends only on the degree sequence. However, the significance is heavily contingent on fixing the proof gaps described below; in its present form, the main theorems are not established.

major comments (3)

[Section 2] The assertion 'All linear bicyclic k-uniform hypergraphs with n vertices and m edges consist of the following two types B_n^k and C_n^k' is made without proof or citation. This classification is load-bearing: Theorems 3.3-3.5 maximize over B_n^k and C_n^k, and if any linear bicyclic hypergraph falls outside these families, the claimed global maximum does not follow. A complete proof of this classification, or a precise reference containing it, must be supplied.
[Lemma 2.1 and Theorems 3.3-3.5] Lemma 2.1 only proves M(H') > M(H); it does not prove that H' is linear, and in fact the edge-moving operation can destroy linearity. For example, in the linear 3-uniform hypercycle with edges e1={a,b,x}, e2={b,c,y}, e3={c,a,z}, moving e2 from b to a gives e2'={a,c,y}, which shares {a,c} with e3, so linearity fails. This is precisely the type of move used repeatedly in the proofs. Consequently, the chains of inequalities in Theorems 3.3-3.5 may compare the original hypergraph with non-linear objects, so the claimed maximum within the linear class is not established. The authors must either prove that each specific move they perform preserves linearity (and membership in B/C), or replace the argument.
[Theorems 3.3 and 3.4] The 'repeating the operation of moving edges' arguments are not rigorous. The paper does not justify that the process terminates at the stated target hypergraph, nor that the hypothesis d_H(v) > d_H(u) - t of Lemma 2.1 holds at every intermediate step. For example, in Theorem 3.3 the statement 'any hypergraph in B^k_{1,n,g,l,q} can be changed into H_1' is asserted without a constructive or inductive proof, and in Theorem 3.4 moves such as 'moving g_l from vertex v to vertex v_2' are declared to produce a hypergraph in C^k_{1,n,p,...} without checking linearity or girth preservation. These gaps are load-bearing because the extremal comparison is built entirely on such moves.

minor comments (4)

[Theorem 3.1] The statement 'the hypergraph with maximum degree 2 has minimum Zagreb index' should read 'any hypergraph with maximum degree 2'; the proof shows all such hypergraphs have the same minimal value, and if none exists the statement is vacuous.
[Section 2] The notation is confusing: B^k_{i,n,p,l,q} is sometimes used for a set of hypergraphs and sometimes for a single hypergraph (e.g., B^k_{1,n,g,0,g}(m-2g)). Please distinguish sets from individual hypergraphs consistently.
[Throughout] There are numerous typographical issues, including 'S3_i=1' instead of a union symbol, inconsistent commas in subscripts, and the phrase 'linear bicyclick' (should be 'bicyclic'). A careful proofreading is needed.
[Theorem 3.3] The formula for M(B^k_{1,n,g,0,g}(m-2g)) is presented without derivation; providing the underlying degree sequence would improve verifiability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the extremal proofs are self-contained algebraic and monotonicity arguments, and the sole self-citation provides background context rather than load-bearing support.

full rationale

The paper defines the Zagreb index as the sum of squared vertex degrees and then derives its extremal results through direct degree-counting identities (Theorem 3.1), an edge-moving inequality (Lemma 2.1) that shows a strict increase in the Zagreb index, and case-by-case comparisons over the explicit families B_n^k and C_n^k with concrete quadratic formulas for M(H) and their monotonicity in the parameters. No parameter is fitted to data and then renamed as a prediction; the target extremal hypergraphs are not used as inputs to their own derivation. The structural assertion in Section 2 that all linear bicyclic k-uniform hypergraphs belong to B_n^k or C_n^k is stated without proof, and Lemma 2.1 is not shown to preserve linearity, but these are gaps or correctness risks, not circularity: they do not make the conclusion equivalent to an input or to a self-citation. The only self-citation, reference [19], is cited for previously known results on hypertrees and linear unicyclic hypergraphs and is not used in the proof of the bicyclic theorems. Accordingly, the paper's derivation chain is self-contained with respect to circularity.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

No free parameters or invented entities. The derivation rests on two unproved structural premises: the B/C classification and preservation of linearity under edge moves. These are the main reasons the proof is incomplete.

assumptions (3)

ad hoc to paper Every connected linear k-uniform hypergraph with n=m(k-1)-1 belongs to B_n^k or C_n^k as defined in Section 2.
Stated without proof or citation before Theorems 3.3 to 3.5; exhaustive classification is load-bearing for the maximum characterizations.
ad hoc to paper The edge-moving operation of Lemma 2.1, when applied repeatedly in the proofs, stays inside the class of linear bicyclic hypergraphs.
Lemma 2.1 proves only the index increase; linearity and connectedness after each move are asserted via 'Obviously' in Theorems 3.3 and 3.4, but not demonstrated. A k=3 example shows the operation can break linearity in general.
domain assumption For the minimum theorem, a linear bicyclic k-uniform hypergraph with all degrees at most 2 exists for the stated n and m.
Theorem 3.1 defines the extremal object by a degree condition without constructing one; existence is plausible from the B_3 construction but not shown.

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Cite this review

Pith. "Pith review of Extremal Zagreb indices of bicyclic hypergraphs." pith.science (2026). https://pith.science/paper/QRS2DNL2

@misc{pith2026250608875,
  author       = {Pith},
  title        = {Pith review of: Extremal Zagreb indices of bicyclic hypergraphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QRS2DNL2}},
  note         = {Machine review of arXiv:2506.08875}
}

abstract

The Zagreb index of a hypergraph is defined as the sum of the squares of the degrees of its vertices. A connected $k$-uniform hypergraph with $n$ vertices and $m$ edges is called bicyclic if $n=m(k-1)-1$. In this paper, we determine the hypergraphs with the maximum and minimum Zagreb indices among all linear bicyclic uniform hypergraphs.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Lexicographical ordering by spectral moments of bicyclic hypergraphs
math.CO 2025-06 conditional novelty 6.0 of 10

For linear k-uniform bicyclic hypergraphs with fixed girth and number of edges, the paper determines which hypergraph is first and which is last in the spectral-moment order.

Reference graph

Works this paper leans on

20 extracted references · 19 canonical work pages · cited by 1 Pith paper

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

Gutman and N

I. Gutman and N. Trinajsti´ c. Graph theory and molecular orbitals. Totalπ- electron energy of alternant hydrocarbons.Chem. Phys. Lett., 17(4):535–538, 1972

work page 1972

[2] [2]

Nikoli´ c, G

S. Nikoli´ c, G. Kovaˇ cevi´ c, A. Miliˇ cevi´ c, and N. Trinajsti´ c. The Zagreb indices 30 years after.Croat. Chem. Acta, 76(2):113–124, 2003

work page 2003

[3] [3]

Gutman and K.C

I. Gutman and K.C. Das. The first Zagreb index 30 years after.MATCH Commun. Math. Comput. Chem, 50:83–92, 2004

work page 2004

[4] [4]

H. Deng. A unified approach to the extremal Zagreb indices for trees, unicyclic graphs and bicyclic graphs.MATCH Commun. Math. Comput. Chem., 57:597– 616, 2007

work page 2007

[5] [5]

Borovi´ canin and T.A

B. Borovi´ canin and T.A. Lampert. On the maximum and minimum Zagreb indices of trees with a given number of vertices of maximum degree.MATCH Commun. Math. Comput. Chem., 74:81–96, 2015. 12

work page 2015

[6] [6]

Borovi´ canin and B

B. Borovi´ canin and B. Furtula. On extremal Zagreb indices of trees with given domination number.Appl. Math. Comput., 279:208–218, 2016

work page 2016

[7] [7]

Pei and X

L. Pei and X. Pan. Extremal values on Zagreb indices of trees with given distancek-domination number.J. Inequal. Appl., 2018:16, 2018

work page 2018

[8] [8]

Zhang and H

S. Zhang and H. Zhang. Unicyclic graphs with the first three smallest and largest first general Zagreb index.MATCH Commun. Math. Comput. Chem., 55:427–438, 2006

work page 2006