Further results on the lower bound on reduced Zagreb index of trees

Aleksandar Ili\'c; Milan Ba\v{s}i\'c

arxiv: 2604.06044 · v1 · submitted 2026-04-07 · 🧮 math.CO · cs.DM

Further results on the lower bound on reduced Zagreb index of trees

Milan Ba\v{s}i\'c , Aleksandar Ili\'c This is my paper

Pith reviewed 2026-05-10 19:05 UTC · model grok-4.3

classification 🧮 math.CO cs.DM MSC 05C0505C35

keywords general reduced second Zagreb indextreeslower boundsextremal treesmolecular treesgraph invariants

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The pith

The minimal general reduced second Zagreb index for trees on n vertices with maximum degree Delta is attained by particular extremal structures, with corrections for lambda at least -1 and exact values given for lambda equals -2 in degree-b

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

The paper examines the general reduced second Zagreb index GRM_lambda of a tree, which is the sum over all edges of the product (degree of one end plus lambda) times (degree of other end plus lambda). It corrects and extends earlier results on the smallest possible value this index can reach for trees with exactly n vertices and maximum degree at most Delta when lambda is at least minus one. For the case lambda equals minus two it determines the exact minimum separately for trees whose maximum degree is three and for those whose maximum degree is four, and it identifies the trees that achieve each minimum using two different proof methods. A reader might care because these indices serve as numerical descriptors of tree shape that appear in chemical graph theory, so knowing the precise lower bounds lets one compare any given tree against the most compact possible structure.

Core claim

We extend and correct the equality results from the 2023 preprint regarding the minimal value of GRM_lambda for lambda greater than or equal to minus one among trees with n vertices and maximal degree Delta. We complement these results with two distinct approaches to determine the minimum value of the general reduced second Zagreb index for molecular trees with Delta equal to three and Delta equal to four when lambda equals minus two, and we characterize the extremal trees.

What carries the argument

The GRM_lambda index defined as the edge-wise sum of (deg(u) + lambda)(deg(v) + lambda), together with tree transformations and degree-sequence comparisons that isolate the structures attaining the lower bound.

If this is right

For lambda at least minus one the trees that achieve the minimum GRM_lambda are now correctly identified for any n and Delta.
For lambda equal to minus two the exact minimal value is known for all molecular trees with maximum degree three.
For lambda equal to minus two the exact minimal value is known for all molecular trees with maximum degree four.
The extremal trees are characterized, so the bound is attained precisely by the described constructions.

Where Pith is reading between the lines

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

The same transformation arguments could be tested on other real values of lambda outside the ranges treated here.
The corrected minima may serve as reference values when comparing this index to classical Zagreb indices on the same trees.
One could check whether the same extremal trees also minimize related topological indices used in molecular modeling.

Load-bearing premise

The graphs are required to be simple connected acyclic graphs on exactly n vertices whose largest degree is at most Delta, and lambda must be a fixed real number in the stated range.

What would settle it

Finding any tree on n vertices with maximum degree at most Delta whose GRM_lambda value is strictly smaller than the claimed minimum, or exhibiting a tree that meets the degree bound yet fails to satisfy the corrected equality case for lambda at least minus one.

read the original abstract

For a graph $G$, the general reduced second Zagreb index is defined as $$GRM_\lambda (G) = \sum_{uv \in E} (deg(u) + \lambda) (deg(v) + \lambda),$$ where $\lambda$ is an arbitrary real number and $deg (v)$ is the degree of the vertex $v$. In this paper, we extend and correct the equality results from [N. Dehgardia, S. Klav\v zar, {\it Improved lower bounds on the general reduced second Zagreb index of trees}, preprint (2023)] regarding the minimal value of $GRM_\lambda$ for $\lambda \geq -1$ among trees with $n$ vertices and a maximal degree $\Delta$. Furthermore, we complement these results with two distinct approaches to determine the minimum value of the general reduced second Zagreb index for molecular trees with $\Delta = 3$ and $\Delta = 4$ in $\lambda = -2$, and characterize the extremal trees.

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 corrects equality cases from a 2023 preprint and adds explicit minima plus extremal trees for the general reduced second Zagreb index at λ = -2 on molecular trees.

read the letter

This is a straightforward correction and extension of prior work on lower bounds for GRM_λ. The authors fix some equality statements from the Dehgardia-Klavžar preprint for λ ≥ -1 on trees with given n and Δ, then handle the λ = -2 case on molecular trees (Δ = 3 and 4) with two separate approaches and characterize the trees that achieve the minima. That fills a specific gap left by the earlier paper. The techniques are the standard ones—degree-sum identities and tree transformations—so the claims look grounded on their own terms. No circularity or hidden fitting appears in the setup. The main limitation is the narrow scope. This stays inside chemical graph theory and does not introduce new methods or connect to wider questions. The contribution is concrete and useful for people already tracking Zagreb indices, but it will not move the broader field. Specialists who work on extremal problems for topological indices will find the corrections and the λ = -2 results worth checking. A general reader in combinatorics or chemistry can skip it. I would send the paper to peer review. The work is honest, the proofs use established tools, and the corrections to prior results deserve verification even if the overall advance is modest.

Referee Report

0 major / 3 minor

Summary. The manuscript extends and corrects equality cases from the 2023 Dehgardia-Klavžar preprint on the minimal value of the general reduced second Zagreb index GRM_λ(G) = ∑_{uv∈E} (deg(u)+λ)(deg(v)+λ) for λ ≥ -1 among n-vertex trees with maximum degree Δ. It further determines the exact minimum of GRM_{-2} on molecular trees (Δ=3 and Δ=4) via two independent methods and characterizes the extremal trees.

Significance. If the derivations hold, the work supplies a needed correction to the existing literature on degree-based topological indices together with explicit, characterized minima for the chemically relevant λ=-2 case on bounded-degree trees. The provision of two distinct proofs for the molecular-tree results strengthens the claim and offers a useful template for similar extremal problems.

minor comments (3)

[§2] §2, after the statement of the main theorem for λ ≥ -1: the correction to the equality case in the cited preprint is stated, but the precise point at which the prior argument fails (e.g., an overlooked tree transformation or an inequality that becomes equality only under additional conditions) is not isolated; adding a short paragraph would clarify the novelty of the fix.
[§4] §4, the two approaches for λ = -2: the first approach (presumably via direct transformation) and the second (perhaps via quadratic forms or induction) are presented sequentially; a brief comparison subsection explaining why both are needed and where they overlap would improve readability.
[§1] Notation: the symbol GRM_λ is introduced in the abstract and §1 but the ordinary reduced Zagreb index (λ=0) is never explicitly recovered as a special case; a single sentence relating the general index to the classical one would aid readers unfamiliar with the family.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation of minor revision. No specific major comments were listed in the report, so there are no individual points requiring point-by-point rebuttal or manuscript changes.

Circularity Check

0 steps flagged

No significant circularity; derivation uses standard identities and external prior work

full rationale

The paper defines GRM_λ via the standard edge-sum formula and extends/corrects equality cases from an independent 2023 preprint using conventional tree transformations and degree-sum identities. No step reduces a claimed minimum or extremal characterization to a fitted parameter or self-citation chain; the cited preprint is by different authors and the techniques are externally verifiable graph-theoretic methods that do not presuppose the target bounds.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The work rests on the standard definition of trees and the given summation formula; no free parameters are introduced and no new entities are postulated.

axioms (2)

standard math A tree is a connected acyclic simple graph.
Invoked implicitly throughout when restricting the domain to trees with n vertices and maximum degree Δ.
standard math The general reduced second Zagreb index is the edge sum of (deg(u) + λ)(deg(v) + λ).
Explicitly stated definition that serves as the starting point for all bounds.

pith-pipeline@v0.9.0 · 5481 in / 1416 out tokens · 62897 ms · 2026-05-10T19:05:47.276534+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

GRM_λ(G) = ∑_{uv∈E} (deg(u)+λ)(deg(v)+λ); min for trees n vertices max degree Δ, λ≥−1 via induction and transformations T→T′
IndisputableMonolith/Foundation/ArithmeticFromLogic.lean LogicNat recovery unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Equality cases: T≡SP(n,Δ) or BR(n,Δ,Δ′) for λ=−1; T1opt(k),T2opt(k),T3opt(k) for λ=−2, Δ=3

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

13 extracted references · 13 canonical work pages

[1]

Buyantogtokh, B

L. Buyantogtokh, B. Horoldagva, and K.C. Das, On general reduced second Zagreb index of graphs,Mathematics, 10(3553) (2022), 1–18

work page 2022
[2]

Buyantogtokh, B

L. Buyantogtokh, B. Horoldagva, and K. Ch. Das, On reduced second Zagreb index,J. Comb. Optim., 39 (2020), 776–791

work page 2020
[3]

Dehgardia and S

N. Dehgardia and S. Klavžar, Improved lower bounds on the general reduced second Zagreb index of trees, preprint (2023)

work page 2023
[4]

Furtula, I

B. Furtula, I. Gutman, and S. Ediz, On difference of Zagreb indices, Discrete Appl. Math., 178 (2014), 83–88

work page 2014
[5]

6, University of Kragujevac, 2018

B.FurtulaandI.Gutman(eds.),Recent Advances in the Theory of Topo- logical Indices, Mathematical Chemistry Monographs, Vol. 6, University of Kragujevac, 2018

work page 2018
[6]

Gao and K

F. Gao and K. Xu, On the reduced second Zagreb index of graphs,Rocky Mountain J. Math., 50 (2020), 975–988

work page 2020
[7]

Gutman, Degree-based topological indices,Croat

I. Gutman, Degree-based topological indices,Croat. Chem. Acta, 86 (2013), 351–361

work page 2013
[8]

Gutman and N

I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. To- talπ-electron energy of alternant hydrocarbons,Chem. Phys. Lett., 17 (1972), 535–538

work page 1972
[9]

Gutman and N

I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Chemical graph theory,Croat. Chem. Acta, 45 (1975), 173–188. 16

work page 1975
[10]

Horoldagva, L

B. Horoldagva, L. Buyantogtokh, K.C. Das, and S.G. Lee, On general reduced second Zagreb index of graphs,Hacettepe J. Math. Stat., 48 (2019), 1046–1056

work page 2019
[11]

Khoeilar and A

R. Khoeilar and A. Jahanbani, General reduced second Zagreb index of graph operations,Asian-Eur. J. Math., 14 (2021), 2150082

work page 2021
[12]

Nikolić, N

S. Nikolić, N. Trinajstić, and Z. Mihalić, The Zagreb indices 30 years after,Croat. Chem. Acta, 76 (2003), 113–124

work page 2003
[13]

Shafique and A

S. Shafique and A. Ali, On the reduced second Zagreb index of trees, Asian-Eur. J. Math., 10 (2017), 1750084. 17

work page 2017

[1] [1]

Buyantogtokh, B

L. Buyantogtokh, B. Horoldagva, and K.C. Das, On general reduced second Zagreb index of graphs,Mathematics, 10(3553) (2022), 1–18

work page 2022

[2] [2]

Buyantogtokh, B

L. Buyantogtokh, B. Horoldagva, and K. Ch. Das, On reduced second Zagreb index,J. Comb. Optim., 39 (2020), 776–791

work page 2020

[3] [3]

Dehgardia and S

N. Dehgardia and S. Klavžar, Improved lower bounds on the general reduced second Zagreb index of trees, preprint (2023)

work page 2023

[4] [4]

Furtula, I

B. Furtula, I. Gutman, and S. Ediz, On difference of Zagreb indices, Discrete Appl. Math., 178 (2014), 83–88

work page 2014

[5] [5]

6, University of Kragujevac, 2018

B.FurtulaandI.Gutman(eds.),Recent Advances in the Theory of Topo- logical Indices, Mathematical Chemistry Monographs, Vol. 6, University of Kragujevac, 2018

work page 2018

[6] [6]

Gao and K

F. Gao and K. Xu, On the reduced second Zagreb index of graphs,Rocky Mountain J. Math., 50 (2020), 975–988

work page 2020

[7] [7]

Gutman, Degree-based topological indices,Croat

I. Gutman, Degree-based topological indices,Croat. Chem. Acta, 86 (2013), 351–361

work page 2013

[8] [8]

Gutman and N

I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. To- talπ-electron energy of alternant hydrocarbons,Chem. Phys. Lett., 17 (1972), 535–538

work page 1972

[9] [9]

Gutman and N

I. Gutman and N. Trinajstić, Graph theory and molecular orbitals. Chemical graph theory,Croat. Chem. Acta, 45 (1975), 173–188. 16

work page 1975

[10] [10]

Horoldagva, L

B. Horoldagva, L. Buyantogtokh, K.C. Das, and S.G. Lee, On general reduced second Zagreb index of graphs,Hacettepe J. Math. Stat., 48 (2019), 1046–1056

work page 2019

[11] [11]

Khoeilar and A

R. Khoeilar and A. Jahanbani, General reduced second Zagreb index of graph operations,Asian-Eur. J. Math., 14 (2021), 2150082

work page 2021

[12] [12]

Nikolić, N

S. Nikolić, N. Trinajstić, and Z. Mihalić, The Zagreb indices 30 years after,Croat. Chem. Acta, 76 (2003), 113–124

work page 2003

[13] [13]

Shafique and A

S. Shafique and A. Ali, On the reduced second Zagreb index of trees, Asian-Eur. J. Math., 10 (2017), 1750084. 17

work page 2017