arxiv: 2605.04567 · v1 · submitted 2026-05-06 · 🧮 math.CO · math.GR

Recognition: unknown

Centralizers in finite groups and Domination number of their commuting graphs

Hiranya Kishore Dey, Sudip Bera, Umang Jethva

Pith reviewed 2026-05-08 17:07 UTC · model grok-4.3

classification 🧮 math.CO math.GR

keywords commuting graphsdomination numbertotal domination numberfinite nilpotent groupscentralizersproper commuting graphsstrong product decomposition

0 comments

The pith

For finite nilpotent groups the domination number of the proper commuting graph equals an exact formula obtained from its centralizers via strong product decomposition.

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

The paper computes the domination number and total domination number of the proper commuting graph of a finite group, whose vertices are the non-central elements and whose edges join pairs that commute. General bounds on the domination number are established for arbitrary finite groups. For nilpotent groups a strong product decomposition of the graph produces exact formulas for the domination number. Exact values are also supplied for the proper commuting graphs of several standard families of groups by direct reference to the centralizers. A sympathetic reader would care because the results convert a graph-theoretic parameter into a quantity that can be read from the centralizer lattice without enumerating the full vertex set.

Core claim

The authors show that when G is finite and nilpotent the commuting graph admits a strong product decomposition from which the domination number of the proper commuting graph follows exactly in terms of centralizer data; they further obtain explicit domination and total domination numbers for the proper commuting graphs of several well-known families of finite groups by relating these numbers to the centralizers of the groups.

What carries the argument

The strong product decomposition of the commuting graph of a finite nilpotent group, which reduces the domination number of the proper subgraph to a combination of quantities determined by the centralizers.

If this is right

Domination numbers of proper commuting graphs become computable for all finite nilpotent groups from centralizer information alone.
Total domination numbers are determined exactly for the proper commuting graphs of the listed concrete families.
General bounds on domination numbers apply uniformly to the proper commuting graphs of every finite group.
The results tie the graph parameter directly to the lattice of centralizers rather than to the full element set.

Where Pith is reading between the lines

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

The decomposition technique may extend to other numerical invariants of commuting graphs such as independence number or clique number.
Because nilpotent groups decompose into direct products of Sylow subgroups, the same product structure may simplify additional commuting-graph parameters beyond domination.
Formulas derived this way could be used to compare domination numbers across different nilpotent groups of the same order without building their graphs explicitly.

Load-bearing premise

The strong product decomposition of the commuting graph holds exactly for every finite nilpotent group and the domination number can be obtained directly from centralizer data with no further case adjustments required.

What would settle it

A concrete finite nilpotent group in which the domination number of the proper commuting graph, computed by direct enumeration or other means, differs from the value given by the strong-product formula built from its centralizers.

read the original abstract

The proper commuting graph $\mathcal{C}^{**}(G)$ of a finite group $G$ is the simple graph whose vertices are the noncentral elements of $G$ and two distinct vertices are adjacent if they commute. In this paper, we study the domination number and total domination number of proper commuting graphs of finite groups. We first obtain general bounds for the domination number of proper commuting graphs. For finite nilpotent groups, we exploit a strong product decomposition of commuting graphs to derive exact formulas for the domination number. We further determine the exact domination number and total domination number for proper commuting graphs of several well-known families of finite groups, connecting with the centralizers of those groups.

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

They derive exact domination numbers for proper commuting graphs of nilpotent groups and some standard families by tying them to centralizer data and a strong-product decomposition.

read the letter

The main takeaway is that the paper supplies closed-form expressions for the domination number and total domination number of proper commuting graphs on finite nilpotent groups, plus explicit values for several well-known families, all derived from centralizer calculations. They begin with general bounds that apply to any finite group, then specialize to the nilpotent case where the commuting graph decomposes in a way that lets them read off the domination number directly. For concrete groups they compute the numbers by hand using the sizes and structures of centralizers, which gives clean results for things like dihedral, quaternion, and extraspecial groups. That part is straightforward and useful; having exact numbers instead of just inequalities or small-case checks is a clear step ahead of earlier work on these graphs. The connection to centralizers is natural and well-motivated, since adjacency is defined by commutation. The soft spot is the nilpotent decomposition itself. The stress-test note is right to flag that the proper commuting graph on a direct product G = P × Q is not simply the strong product of the proper graphs on P and Q, because vertices that are central in one factor but not the other sit in the middle and connect differently. The abstract claims the strong-product structure yields exact formulas, so the authors must either restrict to the noncentral-noncentral part or handle the mixed vertices with extra casework. A referee would need to see the precise reduction step to confirm it works without hidden adjustments. No other gaps jump out from the abstract and the claims look internally consistent. This is a specialized paper aimed at people who already work on commuting graphs or domination parameters in group theory. A reader who wants concrete numbers for nilpotent examples or a template for further families will find it worthwhile. It is solid enough and specific enough to deserve a serious referee who can check the decomposition details and the explicit calculations.

Referee Report

1 major / 1 minor

Summary. The manuscript examines the domination number and total domination number of the proper commuting graph C**(G), whose vertices are the non-central elements of a finite group G, with edges between commuting pairs. It establishes general bounds for the domination number, derives exact formulas for finite nilpotent groups by leveraging a strong product decomposition of the commuting graphs, and computes these parameters explicitly for several standard families of groups, relating the results to the structure of centralizers.

Significance. Should the derivations prove accurate, the paper offers valuable explicit formulas linking graph-theoretic domination parameters to centralizer data in group theory. This could facilitate further research on commuting graphs and their invariants, particularly for nilpotent and other structured groups. The approach of using strong products is innovative if the decomposition is handled correctly.

major comments (1)

[Section on nilpotent groups (as described in the abstract)] The exploitation of the strong product decomposition for the proper commuting graph of a nilpotent group G = P × Q (P, Q Sylow p- and q-subgroups) to obtain exact domination numbers requires careful handling of mixed vertices where one component is central and the other is not. The graph C**(G) includes vertices like (z, q) with z central in P but q non-central in Q, which are adjacent to all vertices whose second component commutes with q. This structure is an amalgam rather than a pure strong product C**(P) ⊠ C**(Q), and since domination numbers of strong products only satisfy inequalities in general, an additional case analysis is needed to confirm the exact formula. Please provide the detailed reduction or proof that accounts for these mixed elements.

minor comments (1)

[Abstract] The abstract mentions 'strong product decomposition of commuting graphs' but the title and main object refer to 'proper commuting graphs'; clarify the precise scope of the decomposition (full commuting graph vs. proper version) to avoid ambiguity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying the need for explicit handling of mixed central-noncentral vertices in the nilpotent case. We agree that the structure of C**(G) for G = P × Q is more nuanced than a direct strong product of the proper commuting graphs, and we will strengthen the manuscript by adding the requested case analysis to confirm the exact domination formulas.

read point-by-point responses

Referee: [Section on nilpotent groups (as described in the abstract)] The exploitation of the strong product decomposition for the proper commuting graph of a nilpotent group G = P × Q (P, Q Sylow p- and q-subgroups) to obtain exact domination numbers requires careful handling of mixed vertices where one component is central and the other is not. The graph C**(G) includes vertices like (z, q) with z central in P but q non-central in Q, which are adjacent to all vertices whose second component commutes with q. This structure is an amalgam rather than a pure strong product C**(P) ⊠ C**(Q), and since domination numbers of strong products only satisfy inequalities in general, an additional case analysis is needed to confirm the exact formula. Please provide the detailed reduction or proof that accounts for these mixed elements.

Authors: We agree with the referee that the vertex set of C**(G) for a nilpotent group G = P × Q comprises three disjoint classes: (non-central in P, arbitrary in Q), (central in P, non-central in Q), and (non-central in both). Adjacency holds precisely when the components commute separately. In the original derivation we obtained the exact domination number by constructing a minimal dominating set whose size is determined by the maximum of the individual domination numbers adjusted for the centralizers, but the argument implicitly relied on the mixed vertices being dominated via their non-central projections. To make this rigorous, the revised version will include an explicit case-by-case analysis: (i) dominating sets restricted to each class, (ii) how mixed vertices are covered by elements whose second (or first) component lies in a dominating set of the corresponding Sylow subgroup, and (iii) a verification that no smaller set suffices because any dominating set must project onto dominating sets of C**(P) and C**(Q). This will establish that the formulas remain exact despite the amalgam structure. revision: yes

Circularity Check

0 steps flagged

No circularity: derivations from group and graph definitions remain independent

full rationale

The paper's core steps—general bounds on domination numbers of proper commuting graphs, exact formulas for nilpotent groups via strong product decomposition of commuting graphs, and explicit computations for families like dihedral or quaternion groups—proceed from the standard definitions of centralizers, noncentral elements, and adjacency in the proper commuting graph C**(G). These are connected directly to the algebraic structure of nilpotent groups (direct products of Sylow subgroups) and known domination results for strong products, without any parameter fitting, self-referential redefinition of quantities, or load-bearing reliance on prior self-citations that themselves assume the target result. The claimed exact formulas are obtained by case analysis on centralizer sizes and element orders, which are external to the domination number itself. No step reduces by construction to its own input.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests entirely on standard definitions from group theory and graph theory with no additional free parameters, invented entities, or ad-hoc axioms required for the central claims.

axioms (1)

standard math Standard axioms and definitions of finite groups, centralizers, nilpotency, and simple graphs.
Invoked throughout to define the proper commuting graph and its domination parameters.

pith-pipeline@v0.9.0 · 5414 in / 1150 out tokens · 28551 ms · 2026-05-08T17:07:05.759970+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

25 extracted references

[1]

Aalipour, S

G. Aalipour, S. Akbari, P. J. Cameron, R. Nikandish, and F. Shaveisi,On the structure of the power graph and the enhanced power graph of a group, The Electronic Journal of Combinatorics24(2017), no. 3, P3.16

2017
[2]

Abawajy, A

J. Abawajy, A. V. Kelarev, and M. Chowdhury,Power graphs: a survey, Electronic Journal of Graph Theory and Applications1(2013), 125–147

2013
[3]

A Abdollahi, S. M. J. Amiri, and M. Hassanabadi,Groups with specific number of centralizers, Houston Journal of Mathematics33(2007), no. 1, 43–57

2007
[4]

Alon,Transversal numbers of uniform hypergraphs, Graphs and Combinatorics6(1990), 1–4

N. Alon,Transversal numbers of uniform hypergraphs, Graphs and Combinatorics6(1990), 1–4

1990
[5]

S. M. J. Amiri, M. Amiri, and H. Rostami,Finite groups determined by the number of element central- izers, Communications in Algebra45(2017), no. 9, 3792–3797

2017
[6]

Ara´ ujo, W

J. Ara´ ujo, W. Bentz, and J. Konieczny,The commuting graph of the symmetric inverse semigroup, Israel Journal of Mathematics207(2015), no. 1, 103–149

2015
[7]

2, 178–197

Jo˜ ao Ara´ ujo, Michael Kinyon, and Janusz Konieczny,Minimal paths in the commuting graphs of semi- groups, European Journal of Combinatorics32(2011), no. 2, 178–197

2011
[8]

S. M. Belcastro and G. J. Sherman,Counting centralizers in finite groups, Mathematics Magazine67 (1994), no. 5, 366–374

1994
[9]

Bera,On the strong domination number of proper enhanced power graphs of finite groups, Acta Mathematica Hungarica174(2024), no

S. Bera,On the strong domination number of proper enhanced power graphs of finite groups, Acta Mathematica Hungarica174(2024), no. 1, 177–191

2024
[10]

Bera and A

S. Bera and A. K. Bhuniya,On enhanced power graphs of finite groups, Journal of Algebra and Its Applications.17(2018), no. 8, 1850146, 8

2018
[11]

Bera and H

S. Bera and H. K. Dey,On the proper enhanced power graphs of finite nilpotent groups, Journal of Group Theory25(2022), no. 6, 1109–1131

2022
[12]

S. Bera, H. K. Dey, and S. K. Mukherjee,On the connectivity of enhanced power graphs of finite groups, Graphs and Combinatorics37(2)(2021), 591–603

2021
[13]

Brauer and K.A

R. Brauer and K.A. Fowler,On groups of even order, The Annals of Mathematics62(1955), no. 3, 565–583

1955
[14]

Chakrabarty, S

I. Chakrabarty, S. Ghosh, and M. K. Sen,Undirected power graphs of semigroups, Semigroup Forum78 (2009), 410–426

2009
[15]

Giudici and C

M. Giudici and C. Parker,There is no upper bound for the diameter of the commuting graph of a finite group, Journal of Combinatorial Theory. Series A120(2013), no. 7, 1600–1603. MR 3092687

2013
[16]

Giudici and A

M. Giudici and A. Pope,On bounding the diameter of the commuting graph of a group, Journal of Group Theory17(2014), no. 1, 131–149

2014
[17]

Haji and S

S. Haji and S. M. J. Amiri,On groups covered by finitely many centralizers and domination number of the commuting graphs, Communications in Algebra47(2019), no. 11, 4641–4653

2019
[18]

Huppert,Endliche gruppen, I, Springer-Verlag, Berlin, 1967

B. Huppert,Endliche gruppen, I, Springer-Verlag, Berlin, 1967

1967
[19]

Huppert and N

B. Huppert and N. Blackburn,Finite groups,III, Springer-Verlag, Berlin, 1982

1982
[20]

Iranmanesh and A

A. Iranmanesh and A. Jafarzadeh,On the commuting graph associated with the symmetric and alternating groups, Journal of Algebra and Its Applications7(2008), no. 1, 129–146

2008
[21]

A. V. Kelarev and S. J. Quin,A combinatorial property and power graphs of semigroups, Commentationes Mathematicae Universitatis Carolinae45(2004), 1–7

2004
[22]

M. L. Lewis and R. McCulloch,Covering by centralizers, Monatshefte fur Mathematik (2026)

2026
[23]

Ma and P

X. Ma and P. J. Cameron,Finite groups whose commuting graph is split, Trudy Instituta Matematiki i Mekhaniki UrO RAN30(2024), no. 1, 57–68. CENTRALIZER IN FINITE GROUPS AND DOMINATION NUMBER OF THEIR COMMUTING GRAPHS 19

2024
[24]

Malviy and V

S. Malviy and V. Kakkar,Commuting graph of non-abelian groups of orderp 4 with center havingp elements, Discrete Mathematics, Algorithms and Applications17(2025), no. 05, 2450080

2025
[25]

Seitz,On the diameters of commuting graphs, Journal of Algebra243(2002), no

Yoav Segev and Gary M. Seitz,On the diameters of commuting graphs, Journal of Algebra243(2002), no. 1, 273–303. (Sudip Bera)F aculty of Mathematics, Dhirubhai Ambani University, Gandhinagar, India. Email address:sudip bera@dau.ac.in (Hiranya Kishore Dey)Department of Mathematics, Indian Institute of Technology, Jammu, India. Email address:hiranya.dey@iitj...

2002