Independent domination polynomial of comaximal graphs of commutative rings

Abstract: The comaximal graph $ Γ(R) $ of a commutative ring $R$ is a simple graph with vertex set $ R $ and two distinct vertices $ a $ and $b $ of $ Γ(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}(Γ(\mathbb{Z}{n}),x) $ of $ Γ(\mathbb{Z}{n}) $ is discussed, along with its unimodal and log-concave properties for certain values of $n$. Some auxiliary results related to $D_{i}(Γ(\mathbb{Z}{n}),x)$ are presented in terms of their zeros. In addition, we determine the independence polynomial $ I(Γ(\mathbb{Z}{n}),x ) $ of $ Γ(\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(Γ(\mathbb{Z}{n}),x ) $ are established, and their log-concave and unimodal properties are examined.

• The paper derives explicit recursive and closed-form formulas for the independent domination and independence polynomials of Z_n graphs.
• It establishes that the coefficients exhibit unimodality and log-concavity under specific prime factor conditions.
• The work connects the algebraic structure of Z_n with combinatorial graph invariants by analyzing root bounds and polynomial behavior.

Independent Domination and Independence Polynomials of Comaximal Graphs of $\mathbb{Z}_n$

Introduction and Context

This paper investigates combinatorial invariants—specifically, the independent domination polynomial and the independence polynomial—of the comaximal graph $\Gamma(R)$ associated to the commutative ring $R = \mathbb{Z}_n$. $\Gamma(R)$ is a graph whose vertices correspond to ring elements, with adjacency defined by the property that the ideals generated by any two vertices sum to the entire ring.

Previous works have established significant connections between algebraic properties of a ring and structural/spectral invariants of $\Gamma(R)$. This article advances that direction by characterizing the polynomials encoding the counts of independent dominating sets and independent sets, emphasizing their unimodal and log-concave behavior, as well as properties of their zeros, for rings $R = \mathbb{Z}_n$ with specific prime power decompositions. The paper both generalizes and extends prior research on domination-type polynomials, particularly within the context of ring graphs where the underlying algebraic structure influences the combinatorics in nontrivial ways.

Structure and Main Results

This work is organized into two central parts: (1) the study of the independent domination polynomial $D_i(\Gamma(\mathbb{Z}_n), x)$, and (2) the study of the independence polynomial $I(\Gamma(\mathbb{Z}_n), x)$, with emphasis on explicit formulas for specific $n$ and analysis of generic structure for arbitrary $n$.

Comaximal Graph Structure

Fundamental to the analysis is the decomposition of $\Gamma(R)$0 via the partition of $\Gamma(R)$1 according to the greatest common divisor with $\Gamma(R)$2. The graph is canonically expressed as:

$\Gamma(R)$3

where $\Gamma(R)$4 corresponds to the unit elements (each adjacent to all other vertices), $\Gamma(R)$5 is the zero vertex, and $\Gamma(R)$6 encodes the graph induced by non-unit, nonzero-divisorial elements corresponding to proper divisors of $\Gamma(R)$7, with adjacency based on the coprimality of divisors.

Independent Domination Polynomial

The paper provides recursive and explicit expressions for $\Gamma(R)$8 in many cases:

• Prime powers: For $\Gamma(R)$9, $R = \mathbb{Z}_n$0, decomposing the independent dominating sets into those containing one unit or the "divisor class" sets.
• Products of distinct primes: For $R = \mathbb{Z}_n$1, $R = \mathbb{Z}_n$2; for $R = \mathbb{Z}_n$3, an explicit multinomial expression is derived analogously.
• General Prime Decompositions: For $R = \mathbb{Z}_n$4 with $R = \mathbb{Z}_n$5, explicit formulas are given for the polynomial, partitioning independent dominating sets according to subset systems in $R = \mathbb{Z}_n$6.

The analysis extends to the roots of $R = \mathbb{Z}_n$7, showing that real roots are generally confined to $R = \mathbb{Z}_n$8 for the majority of cases, with nontrivial complex roots emerging as the exponents and the number of primes increase.

Unimodality and Log-concavity

The coefficients of $R = \mathbb{Z}_n$9 are analyzed with respect to unimodality and log-concavity:

• Unimodality: For $\Gamma(R)$0, $\Gamma(R)$1 is unimodal if and only if $\Gamma(R)$2 or $\Gamma(R)$3; for biprime powers, similar sharp conditions are established.
• Log-concavity: These are fully characterized with sharp conditions on the prime decompositions, highlighting exceptions (such as $\Gamma(R)$4) where log-concavity fails due to structural zeros in the coefficient sequence.
• The existence of gaps ("oscillations") between nonzero coefficients is shown to precisely delimit the possible unimodal and log-concave regimes.

Independence Polynomial

The independence polynomial $\Gamma(R)$5 is determined explicitly for the principal cases:

• For $\Gamma(R)$6, $\Gamma(R)$7.
• For $\Gamma(R)$8,

$\Gamma(R)$9

• For $\Gamma(R)$0, $\Gamma(R)$1, reflecting disjoint union of the units/zero and remaining sets.
• For biprime powers, multinomial expressions are obtained.

The authors exploit the Eneström-Kakeya theorem to bound the nontrivial roots of $\Gamma(R)$2 in the complex plane, linking the bounds directly to prime factors in the ring's structure.

The unimodality and log-concavity of $\Gamma(R)$3 are characterized. The independence polynomial may not be unimodal or log-concave for multi-prime rings, and this is demonstrated through both combinatorial and algebraic reasoning for chosen parameters. Explicit examples are given for $\Gamma(R)$4 and $\Gamma(R)$5, with the location of their zeros provided.

Implications and Theoretical Perspectives

This analysis systematically connects the algebraic structure of finite commutative rings $\Gamma(R)$6 with the combinatorial invariants of their comaximal graphs, demonstrating that the decomposition of $\Gamma(R)$7 sharply determines the behavior of both the independent domination and independence polynomials.

From a combinatorics and algebraic graph theory standpoint, these results clarify:

• How the factorization of $\Gamma(R)$8 into primes (and the distribution of exponents) predicts combinatorial features such as the number, sizes, and arrangement of independent dominating sets.
• The spectral and algebraic properties of associated graph polynomials, such as unimodality/log-concavity, and how algebraic restrictions on $\Gamma(R)$9 yield precise necessary and sufficient conditions for these properties.
• Constraints on the location of roots of independence polynomials, yielding both analytic and geometric insights into the coefficients and combinatorial classes.

Practically, these results have implications for any context where domination-type invariants are linked to underlying algebraic data structures—e.g., error-correcting codes, network design in algebraically structured settings, and computational approaches to NP-complete domination/invariant counting problems.

Theoretically, open problems remain regarding the general structure of $R = \mathbb{Z}_n$0 for arbitrary $R = \mathbb{Z}_n$1, extension to more general classes of commutative rings, and the limiting behavior and geometry of polynomial roots. The connection between independence/independent domination polynomials and further ring-theoretic invariants (such as ideal-theoretic or homological data) is also an avenue for future research.

Conclusion

This paper establishes a framework for understanding and computing the independent domination and independence polynomials of comaximal graphs for $R = \mathbb{Z}_n$2, providing explicit formulas for key classes of $R = \mathbb{Z}_n$3, complete characterization of unimodality/log-concavity regimes, and analytic bounds for polynomial roots. The results tightly couple algebraic and combinatorial perspectives, and the methodologies presented can be extended to more complex ring structures and related graph polynomials, with numerous open questions remaining concerning the deeper combinatorial-algebraic interplay and its applications.