Independent domination polynomial of comaximal graphs of commutative rings

Bilal Ahmad Rather

arxiv: 2604.03966 · v1 · submitted 2026-04-05 · 🧮 math.CO · cs.DM· math.RA

Independent domination polynomial of comaximal graphs of commutative rings

Bilal Ahmad Rather This is my paper

Pith reviewed 2026-05-13 17:36 UTC · model grok-4.3

classification 🧮 math.CO cs.DMmath.RA

keywords gammamathbbpolynomialcomaximalcommutativedominationgraphindependent

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The pith

The independent domination polynomial D_i(Γ(Z_n), x) is computed for certain n together with its unimodal, log-concave, and zero properties; the independence polynomial I(Γ(Z_n), x) receives similar treatment.

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

Comaximal graphs have vertices as the elements of Z_n, with an edge between a and b when the ideals they generate together equal the whole ring. The independent domination polynomial is the generating function whose coefficients count the independent dominating sets of each possible size. The authors obtain closed-form or recursive expressions for this polynomial when n takes special forms, then verify that the coefficient sequences are unimodal and log-concave and locate the roots. Parallel results are given for the ordinary independence polynomial that simply counts all independent sets.

Core claim

We determine the independence polynomial I(Γ(Z_n),x) of Γ(Z_n) for special values of n and provide a general result associated with it. The bounds for the zero of the polynomial I(Γ(Z_n),x) are established, and their log-concave and unimodal properties are examined.

Load-bearing premise

That the structure of Γ(Z_n) for the chosen n permits closed-form expressions without additional case distinctions or computational search that would invalidate the claimed general result.

read the original abstract

The comaximal graph $ \Gamma(R) $ of a commutative ring $R$ is a simple graph with vertex set $ R $ and two distinct vertices $ a $ and $b $ of $ \Gamma(R) $ are adjacent if and only if $ aR+bR=R $, where $ aR $ is the ideal generated by $ a $ in $ R $. In this article, the independent domination polynomial $ D_{i}(\Gamma(\mathbb{Z}_{n}),x) $ of $ \Gamma(\mathbb{Z}_{n}) $ is discussed, along with its unimodal and log-concave properties for certain values of $n$. Some auxiliary results related to $D_{i}(\Gamma(\mathbb{Z}_{n}),x)$ are presented in terms of their zeros. In addition, we determine the independence polynomial $ I(\Gamma(\mathbb{Z}_{n}),x ) $ of $ \Gamma(\mathbb{Z}_{n}) $ for special values of $n$ and provide a general result associated with it. The bounds for the zero of the polynomial $ I(\Gamma(\mathbb{Z}_{n}),x ) $ are established, and their log-concave and unimodal properties are examined.

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 computes explicit forms for the independence polynomial and independent domination polynomial of the comaximal graph on Z_n for selected n, plus root bounds and checks for unimodality and log-concavity.

read the letter

The main takeaway is that the authors derive closed-form expressions for I(Γ(Z_n), x) and D_i(Γ(Z_n), x) when n takes certain values, then establish bounds on the zeros and confirm the polynomials are unimodal and log-concave in their coefficients. They work from the standard comaximal adjacency rule where edges exist precisely when the generators are coprime in the ring sense. The calculations rely on the natural splitting of the vertex set according to the divisors of n, which keeps the counting manageable for the chosen cases. They also supply some auxiliary facts about the roots that follow from basic polynomial analysis. What the paper does cleanly is turn the graph structure into concrete polynomials without introducing extra parameters or fitting. Once the adjacency is fixed, the independent sets and dominating sets decompose over the prime-power factors, so the explicit formulas are a natural next step from earlier work on these graphs. The property checks are standard applications of known criteria for log-concavity and unimodality. The soft spot is the limit to special n. The claimed general result still appears to rest on case analysis tied to the divisor lattice, which is fine but means the result is not a single formula that covers every n without further splitting. If the proofs handle the counting without hidden restrictions on n, that is acceptable; otherwise the scope stays narrow. This work is for readers already familiar with domination polynomials on algebraic graphs. A specialist in combinatorial ring theory or graph polynomials will find the explicit expressions useful for examples or as a base for further calculations. It deserves a serious referee because the claims are concrete and the derivations are in principle checkable against the graph definition. I would send it to peer review.

Referee Report

0 major / 3 minor

Summary. The manuscript computes the independent domination polynomial D_i(Γ(Z_n),x) of the comaximal graph Γ(Z_n) for selected n, supplies auxiliary results on its zeros, and verifies unimodality and log-concavity. It also derives the independence polynomial I(Γ(Z_n),x) for special n together with a general result, establishes bounds on the zeros of I, and checks the same unimodal and log-concave properties.

Significance. If the explicit formulas and property verifications are correct, the work supplies concrete, usable expressions for two standard graph polynomials on the family of comaximal graphs of Z_n. Such closed-form results on algebraically defined graphs are of interest in algebraic graph theory and can serve as test cases for broader conjectures on unimodality and log-concavity.

minor comments (3)

The abstract distinguishes 'special values of n' from 'a general result' for I(Γ(Z_n),x) without stating the precise condition on n; this distinction should be made explicit in the introduction or the statement of the main theorem.
The manuscript should include a short table or list of the specific n for which closed forms are given, together with the corresponding polynomials, to allow immediate verification.
Notation for the independent domination polynomial is introduced as D_i but later references occasionally use D; consistent use of a single symbol is needed.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript on the independent domination polynomial and independence polynomial of comaximal graphs Γ(Z_n). The assessment that the explicit formulas and property verifications are of interest in algebraic graph theory is appreciated. As no specific major comments were provided in the report, we have no revisions to make at this stage but remain available for any further clarifications the editor may require.

Circularity Check

0 steps flagged

No significant circularity; derivations are direct and self-contained

full rationale

The paper computes the independence polynomial I(Γ(Z_n),x) and independent domination polynomial D_i(Γ(Z_n),x) via explicit case analysis on the divisor lattice of n, using the standard adjacency rule gcd(a,b,n)=1. No equation reduces a claimed prediction or general result to a fitted parameter or self-referential definition. Log-concavity, unimodality, and zero bounds follow from standard polynomial techniques applied to the explicitly derived expressions. No load-bearing self-citation chain or ansatz smuggling is present; the work is self-contained against the ring and graph definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard definition of the comaximal graph via ideal generation in Z_n and the ordinary definitions of independent dominating sets; no new free parameters, ad-hoc axioms, or invented entities are introduced.

axioms (2)

standard math Standard properties of commutative rings Z_n and the ideal generated by an element
Used to define adjacency: a and b are adjacent iff aR + bR = R.
standard math Standard graph-theoretic definitions of independent sets and dominating sets
Basis for the independent domination polynomial and independence polynomial.

pith-pipeline@v0.9.0 · 5509 in / 1232 out tokens · 36683 ms · 2026-05-13T17:36:40.422912+00:00 · methodology

Independent domination polynomial of comaximal graphs of commutative rings

Core claim

Load-bearing premise

discussion (0)