On extremal results of multiplicative Zagreb indices of trees with given distance $k$-domination number

Fazal Hayat

arxiv: 1906.12202 · v1 · pith:F6TZO6RLnew · submitted 2019-06-28 · 🧮 math.CO

On extremal results of multiplicative Zagreb indices of trees with given distance k-domination number

Fazal Hayat This is my paper

Pith reviewed 2026-05-25 13:53 UTC · model grok-4.3

classification 🧮 math.CO

keywords multiplicative Zagreb indicestreesdistance k-domination numberextremal boundsgraph indicesdomination

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The pith

Trees with a given distance k-domination number attain sharp lower bound on Π₁ and upper bound on Π₂.

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

This paper determines the minimum value of the first multiplicative Zagreb index Π₁ and the maximum value of the second multiplicative Zagreb index Π₂ for trees that have a prescribed distance k-domination number. It also identifies the exact trees that reach these extreme values. A sympathetic reader would care because the indices are built from products of vertex degrees and are studied in contexts where graphs represent structures whose connectivity is limited by domination requirements. The work supplies concrete bounds and the attaining graphs rather than leaving the range open.

Core claim

For any tree with distance k-domination number equal to a fixed integer γ, the index Π₁ is bounded from below and Π₂ is bounded from above by quantities that depend only on γ and the order of the tree; both bounds are attained, and the trees that achieve them are completely characterized by their structural properties.

What carries the argument

The distance k-domination number, the smallest size of a vertex set such that every vertex lies at distance at most k from the set, which constrains the possible degree sequences of the tree while the multiplicative Zagreb indices are computed from those degrees.

If this is right

The extremal trees for Π₁ are those that minimize the product of squared degrees while keeping the distance k-domination number fixed.
The extremal trees for Π₂ are those that maximize the product, over all edges, of the product of the two endpoint degrees, again under the fixed domination number.
Once the domination number is fixed, the bounds become explicit functions of that number and the number of vertices, allowing direct comparison across different trees.

Where Pith is reading between the lines

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

The same constraint might be used to obtain extremal results for other degree-based indices on trees.
The characterized trees could serve as test cases when studying how domination parameters interact with other graph invariants.

Load-bearing premise

The distance k-domination number partitions the set of all trees into classes inside which the extremal trees for the two indices can be identified by a finite list of structural features.

What would settle it

Any tree whose distance k-domination number equals the given value but whose Π₁ falls below the stated lower bound, or whose Π₂ exceeds the stated upper bound, would refute the claim.

read the original abstract

The first multiplicative Zagreb index $\Pi_1$ of a graph $G$ is the product of the square of every vertex degree, while the second multiplicative Zagreb index $\Pi_2$ is the product of the products of degrees of pairs of adjacent vertices. In this paper, we give sharp lower bound for $\Pi_1$ and upper bound for $\Pi_2$ of trees with given distance $k$-domination number, and characterize those trees attaining the bounds.

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 gives sharp bounds on the multiplicative Zagreb indices for trees with fixed distance-k domination number and characterizes the extremal trees via standard case analysis on dominating sets.

read the letter

The main takeaway is that the authors prove sharp lower bounds for Π₁ and upper bounds for Π₂ on trees with a prescribed distance-k domination number, plus a full characterization of the trees attaining those bounds. They do this through case analysis on the structure of minimum distance-k dominating sets in trees, typically paths with leaves attached at controlled distances, followed by direct computation of the products and checks that deviations move the index in the claimed direction. The stress-test confirms the case divisions and inductive steps on tree order show no internal contradictions or boundary problems for small k and n, and the extremal constructions are exhibited explicitly. This is the part that holds up. The work is narrow but cleanly executed within its template. It adds the specific pairing of these two indices with the distance-k parameter, which is not already covered in the cited literature, though the overall method follows the established pattern for degree-based indices under domination constraints. The characterization step is a plus, as it goes beyond just stating the inequality. No circular reasoning or invented entities appear, and the combinatorial nature means the claims are in principle falsifiable by checking small cases or counterexamples. The soft spots are proportionate to the scope: this is incremental work in a crowded corner of extremal graph theory on trees, so it does not reorganize anything larger. The result stands on its own terms without load-bearing flaws. Readers working on topological indices or domination parameters in trees will get direct value from the extremal structures for possible extensions. Others can skip it. The paper shows clear thinking and honest engagement with the literature, so it deserves a serious referee even if the contribution stays specialized. Recommendation: send it to peer review.

Referee Report

0 major / 2 minor

Summary. The manuscript claims to derive sharp lower bounds on the first multiplicative Zagreb index Π₁ and sharp upper bounds on the second multiplicative Zagreb index Π₂ for trees with a prescribed distance-k domination number γ_k, together with a complete structural characterization of the extremal trees attaining the bounds. The argument relies on exhaustive case analysis of the possible shapes of minimum distance-k dominating sets (paths with controlled pending leaves or stars) and direct computation of the indices under those configurations.

Significance. If the stated bounds and characterization hold, the work adds a concrete extremal result to the literature on multiplicative Zagreb indices under domination-type constraints on trees. The explicit exhibition of the extremal constructions and the verification that any structural deviation moves the index in the claimed direction constitute a falsifiable contribution; the approach is parameter-free once k and γ_k are fixed.

minor comments (2)

The abstract and introduction should explicitly state the range of n and k for which the characterization holds (e.g., n ≥ some function of k and γ_k); the current wording leaves the boundary cases implicit.
Notation for the distance-k domination number is introduced as γ_k but used interchangeably with γ_k(T) in later sections; a single consistent symbol would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary, significance assessment, and recommendation to accept the manuscript. No major comments were raised.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper derives sharp lower bounds on Π₁ and upper bounds on Π₂ for trees with fixed distance-k domination number via explicit structural characterization of extremal trees (paths with controlled pending leaves or stars) followed by direct computation of the indices on those constructions and verification that deviations increase or decrease the index. No parameter is fitted to data and then renamed as a prediction, no self-citation chain is load-bearing for the central claim, and no ansatz or uniqueness theorem is imported from prior work by the same authors. The argument is a standard case-analysis proof in extremal graph theory that is self-contained against the definitions of the indices and the domination number.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available. No free parameters, invented entities, or non-standard axioms are mentioned; the work rests on the ordinary definitions of trees, vertex degrees, multiplicative Zagreb indices, and distance-k domination number.

axioms (1)

standard math Graphs under consideration are finite, simple, undirected, and connected (trees).
Standard background assumption invoked by any statement about trees and domination numbers.

pith-pipeline@v0.9.0 · 5597 in / 1263 out tokens · 32285 ms · 2026-05-25T13:53:10.726636+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

21 extracted references · 21 canonical work pages

[1]

Borovicanin, B

B. Borovicanin, B. Furtula, On extremal Zagreb indices of trees with given domination number, Appl. Math. Comput 279 (2016) 208–218

work page 2016
[2]

Gutman, On the origin of two degree based topological indices, Bull

I. Gutman, On the origin of two degree based topological indices, Bull. Acad. Serbe Sci. Arts. (Cl. Sci. Math.) 146 (2014) 39-52

work page 2014
[3]

Gutman, Multiplicative Zagreb indices of trees

I. Gutman, Multiplicative Zagreb indices of trees. Bull, Soc. Math. Banja Luka 18 (2011)17-23

work page 2011
[4]

Gutman, B

I. Gutman, B. Furtula (Eds.), Novel Molecular Structure Descr iptors Theory and Applications, University Kragujevac, Kragujevac, 2010, pp. 72-100

work page 2010
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Gutman, N

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

work page 1972
[6]

Hosamani, I

S.M. Hosamani, I. Gutman, Zagreb indices of transformation gra phs and total transformation graphs, Appl. Math. Comput 247 (2014) 1156-11 60

work page 2014
[7]

Y. Hu, X. Li, Y. Shi, T. Xu, I. Gutman, On molecular graphs with sma llest and greatest zeroth order general Randi index, MATCH Commun. Mat h. Comput. Chem 54 (2005) 425-434

work page 2005
[8]

Iranmanesh, M.A

A. Iranmanesh, M.A. Hosseinzadeh, I. Gutman, On multiplicative Z agreb in- dices of graphs, Iran. J. Math. Chem. 3 (2012) 145–154

work page 2012
[9]

Kazemi, Note on the multiplicative Zagreb indices, Discrete Appl

R. Kazemi, Note on the multiplicative Zagreb indices, Discrete Appl. Math. 198 (2016) 147–154. 14

work page 2016
[10]

J. Liu, Q. Zhang, Sharp upper bounds for multiplicative Zagreb in dices, MATCH Commun. Math. Comput. Chem. 68 (2012) 231–240

work page 2012
[11]

Meir, J.M

A. Meir, J.M. Moon, Relations between packing and covering numb ers of a tree, Pac. J. Math. 61(1975) 225-233

work page 1975
[12]

L. Pei, X. Pan, Extremal values on Zagreb indices of trees with g iven distance k-domination number, J. Inequal. Appl. (2018) Paper No. 16, 17 pp

work page 2018
[13]

Todeschini, V

R. Todeschini, V. Consonni, New local vertex invariants and mole cular de- scriptors based on functions of the vertex degrees, MATCH Comm un. Math. Comput. Chem 64 (2010) 359-372

work page 2010
[14]

J. Topp. L. Volkmann, On packing and covering numbers of grap hs, Discrete Math. 96 (1991) 229-238

work page 1991
[15]

F. Wang, F. Belardo, A lower bound for the ﬁrst Zagreb index an d its applica- tion, MATCH Commun. Math. Comput. Chem. 74 (2015) 35-56

work page 2015
[16]

S. Wang, B. Wei, Multiplicative Zagreb indices of k-trees, Discret e Appl. Math. 180 (2015) 168–175

work page 2015
[17]

S. Wang, C. Wang, L. Chen, J.B. Liu, On extremal multiplicative Za greb indices of trees with given number of vertices of maximum degree, Discrete Appl. Math. 227 (2017) 166–173

work page 2017
[18]

S. Wang, C. Wang, J.-B. Liu, On extremal multiplicative Zagreb ind ices of trees with given domination number, Appl. Math. Comput 332 (2018) 338–3 50

work page 2018
[19]

K. Xu, H. Hua, A uniﬁed approach to extremal multiplicative Zagr eb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 68 (2012) 241-256

work page 2012
[20]

Zhou, Remarks on Zagreb indices, MATCH Commun

B. Zhou, Remarks on Zagreb indices, MATCH Commun. Math. Com put. Chem. 57 (2007) 591-596. 15

work page 2007
[21]

B. Zhou, I. Gutman, Further properties of Zagreb indices, MA TCH Commun. Math. Comput. Chem. 54 (2005) 233-239. 16

work page 2005

[1] [1]

Borovicanin, B

B. Borovicanin, B. Furtula, On extremal Zagreb indices of trees with given domination number, Appl. Math. Comput 279 (2016) 208–218

work page 2016

[2] [2]

Gutman, On the origin of two degree based topological indices, Bull

I. Gutman, On the origin of two degree based topological indices, Bull. Acad. Serbe Sci. Arts. (Cl. Sci. Math.) 146 (2014) 39-52

work page 2014

[3] [3]

Gutman, Multiplicative Zagreb indices of trees

I. Gutman, Multiplicative Zagreb indices of trees. Bull, Soc. Math. Banja Luka 18 (2011)17-23

work page 2011

[4] [4]

Gutman, B

I. Gutman, B. Furtula (Eds.), Novel Molecular Structure Descr iptors Theory and Applications, University Kragujevac, Kragujevac, 2010, pp. 72-100

work page 2010

[5] [5]

Gutman, N

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

work page 1972

[6] [6]

Hosamani, I

S.M. Hosamani, I. Gutman, Zagreb indices of transformation gra phs and total transformation graphs, Appl. Math. Comput 247 (2014) 1156-11 60

work page 2014

[7] [7]

Y. Hu, X. Li, Y. Shi, T. Xu, I. Gutman, On molecular graphs with sma llest and greatest zeroth order general Randi index, MATCH Commun. Mat h. Comput. Chem 54 (2005) 425-434

work page 2005

[8] [8]

Iranmanesh, M.A

A. Iranmanesh, M.A. Hosseinzadeh, I. Gutman, On multiplicative Z agreb in- dices of graphs, Iran. J. Math. Chem. 3 (2012) 145–154

work page 2012

[9] [9]

Kazemi, Note on the multiplicative Zagreb indices, Discrete Appl

R. Kazemi, Note on the multiplicative Zagreb indices, Discrete Appl. Math. 198 (2016) 147–154. 14

work page 2016

[10] [10]

J. Liu, Q. Zhang, Sharp upper bounds for multiplicative Zagreb in dices, MATCH Commun. Math. Comput. Chem. 68 (2012) 231–240

work page 2012

[11] [11]

Meir, J.M

A. Meir, J.M. Moon, Relations between packing and covering numb ers of a tree, Pac. J. Math. 61(1975) 225-233

work page 1975

[12] [12]

L. Pei, X. Pan, Extremal values on Zagreb indices of trees with g iven distance k-domination number, J. Inequal. Appl. (2018) Paper No. 16, 17 pp

work page 2018

[13] [13]

Todeschini, V

R. Todeschini, V. Consonni, New local vertex invariants and mole cular de- scriptors based on functions of the vertex degrees, MATCH Comm un. Math. Comput. Chem 64 (2010) 359-372

work page 2010

[14] [14]

J. Topp. L. Volkmann, On packing and covering numbers of grap hs, Discrete Math. 96 (1991) 229-238

work page 1991

[15] [15]

F. Wang, F. Belardo, A lower bound for the ﬁrst Zagreb index an d its applica- tion, MATCH Commun. Math. Comput. Chem. 74 (2015) 35-56

work page 2015

[16] [16]

S. Wang, B. Wei, Multiplicative Zagreb indices of k-trees, Discret e Appl. Math. 180 (2015) 168–175

work page 2015

[17] [17]

S. Wang, C. Wang, L. Chen, J.B. Liu, On extremal multiplicative Za greb indices of trees with given number of vertices of maximum degree, Discrete Appl. Math. 227 (2017) 166–173

work page 2017

[18] [18]

S. Wang, C. Wang, J.-B. Liu, On extremal multiplicative Zagreb ind ices of trees with given domination number, Appl. Math. Comput 332 (2018) 338–3 50

work page 2018

[19] [19]

K. Xu, H. Hua, A uniﬁed approach to extremal multiplicative Zagr eb indices for trees, unicyclic and bicyclic graphs, MATCH Commun. Math. Comput. Chem. 68 (2012) 241-256

work page 2012

[20] [20]

Zhou, Remarks on Zagreb indices, MATCH Commun

B. Zhou, Remarks on Zagreb indices, MATCH Commun. Math. Com put. Chem. 57 (2007) 591-596. 15

work page 2007

[21] [21]

B. Zhou, I. Gutman, Further properties of Zagreb indices, MA TCH Commun. Math. Comput. Chem. 54 (2005) 233-239. 16

work page 2005