Clique-Cover Polynomials in Graph Theory

• Clique-cover polynomials are generating functions that count vertex-disjoint clique partitions in graphs, connecting combinatorial enumeration with algebraic properties.
• They establish a duality with chromatic polynomials using Stirling transforms, while their adjoint forms correlate with the independence polynomials of auxiliary graphs.
• These polynomials enable efficient algorithmic computation for cographs and foster analyses of real-rootedness, log-concavity, and other key combinatorial features in graph theory.

A clique-cover polynomial is a generating function encoding the number of ways a simple graph can be partitioned into vertex-disjoint cliques. Closely related to the chromatic and matching polynomials, clique-cover polynomials and their alternated-sign forms—the adjoint polynomials—exhibit deep duality, transformation, and root-structure properties, and are connected to independence polynomials of derived auxiliary graphs. These objects facilitate both combinatorial enumeration and algebraic analysis of graph coverings, connecting diverse areas within algebraic and enumerative graph theory (Jumadildayev, 17 Dec 2025, Bencs, 2017).

1. Definitions and Fundamental Properties

Let $G=(V,E)$ be a simple graph on $n$ vertices. A $k$-clique cover of $G$ is a partition of $V$ into $k$ vertex-disjoint cliques (a singleton is allowed as a trivial clique). If $s_k(G)$ denotes the number of $k$-clique covers, the clique-cover polynomial is

$\xi_G(t) = \sum_{k=1}^n s_k(G)\, t^k,$

with $t$ an indeterminate tracking the number of cliques. This object is also known as the sigma polynomial or, in different normalization, the adjoint polynomial, defined as

$h(G,x) = \sum_{k=1}^n (-1)^{n-k} a_k(G)\, x^k,$

where $a_k(G)$ is the number of $k$-clique covers of $G$ (Bencs, 2017). Adjoint polynomials contain equivalent combinatorial data and are minorants of the chromatic polynomial of the graph complement, with

$$\chi(\overline G,x) = \sum_{k=1}^n a_k(G)\, x(x-1)\cdots(x-k+1).$$

The coefficients $a_k(G)$ serve as the "Stirling-type" building blocks in this expansion.

2. Duality with Chromatic Polynomials and Change-of-Basis

A central duality interrelates clique-cover polynomials of $G$ and chromatic polynomials of $\overline G$. Birkhoff's inclusion-exclusion yields

$\chi_G(t) = \sum_{k=1}^n s_k(\overline G)\, (t)_k,$

where $(t)_k = t(t-1)\cdots (t-k+1)$ is the falling factorial. Defining a linear operator $\varphi_{\chi}$ with $\varphi_{\chi}[t^n] = (t)_n$ and its inverse $\varphi_\xi$ (which acts as a basis change via the Stirling numbers of the second kind: $\varphi_\xi[t^n] = \sum_{k=1}^n S(n,k) t^k$), one obtains the duality relations: $$\xi_G(t) = \varphi_\xi\big[\,\chi_{\overline G}(t)\,\big], \qquad \chi_G(t) = \varphi_\chi\big[\,\xi_{\overline G}(t)\,\big].$$ Pictorially, this translates proper colorings of $\overline G$ into partitions of $V$—which is exactly a clique cover of $G$ (Jumadildayev, 17 Dec 2025).

3. Transformations: Adjoint and Independence Polynomials

The adjoint polynomial $h(G,x)$ is equivalently described in terms of the independence polynomial of a derived auxiliary graph $\widehat G$. For a fixed ordering of $V(G) = \{u_1,\ldots,u_n\}$, the vertex set of $\widehat G$ is $E(G)$. Two distinct edges $(u_i,u_j)$ and $(u_k,u_\ell)$ are adjacent in $\widehat G$ if any of the following hold: $i = k$, $j = k$, or $j = \ell$ and $(u_i,u_k) \notin E(G)$ (with $j \le \ell$ assumed). Then

$$h^*(G,x) := x^n h\bigl(G,1/x\bigr) = I(\widehat G, x),$$

where $I(\widehat G, x)$ is the independence polynomial of $\widehat G$. There is a bijection between $k$-clique covers of $G$ and independent sets of size $n-k$ in $\widehat G$. The matching polynomial

$M(G,x) = x^n I(L(G), 1/x)$

is structurally analogous, with $L(G)$ the line graph of $G$ (Bencs, 2017).

4. Poisson–Expectation and Analytic Representations

Evaluating the clique-cover polynomial at $\lambda$ admits a probabilistic interpretation: $$\xi_G(\lambda) = \mathbb{E}\big[\,\chi_{\overline G}(X)\,\bigr], \quad X \sim \mathrm{Pois}(\lambda),$$ since the operator $\varphi_\xi$ is the moment-to-ordinary basis change for the Poisson distribution ($\mathbb{E}[(X)_k] = \lambda^k$). This provides a Poisson-integral viewpoint, tightly analogous to Lebesgue–Stieltjes representations used for matching and path-cover polynomials with Gaussian or gamma measures (Jumadildayev, 17 Dec 2025). The analytic properties of clique-cover and adjoint polynomials are thus closely linked to those of independence polynomials.

5. Closed Forms for Special Graph Classes

For complete multipartite graphs $K_{\alpha_1,\ldots,\alpha_m}$ (formed by the join of independent sets), iterating the join-graph formula yields

$$\xi_{K_{\alpha_1,\ldots,\alpha_m}}(t) = \varphi_\xi \left[ \prod_{i=1}^m (t)_{\alpha_i} \right] = \sum_{k=1}^N S(N,k) \left[ t^k \right] \left( \prod_{i=1}^m (t)_{\alpha_i} \right) t^k,$$

where $N = \alpha_1 + \cdots + \alpha_m$. For bipartite complete graphs $K_{a,b}$, two forms are given: $$\xi_{K_{a,b}}(t) = \sum_{i=0}^{\min(a,b)} (-1)^{a+b-(a+b-2i)} \binom{a}{i} \binom{b}{i} t^{a+b-2i} = \sum_{k=1}^{a+b} \left(\sum_{i+j=k} S(a,i)S(b,j)\right) t^k.$$ In the case of $K_n$, $a_k(K_n) = S(n,k)$ and both clique-cover and adjoint polynomials are determined by Stirling numbers (Jumadildayev, 17 Dec 2025, Bencs, 2017).

6. Algorithmic Computation and Cographs

Cographs, constructed by recursive unions and joins from single vertices, allow for efficient computation of clique-cover polynomials. Their cotree representation, with internal nodes labeled as "union" or "join," enables a uniform bottom-up algorithm:

• For leaves, return $t$.
• For union-nodes, take the product of child polynomials.
• For join-nodes, apply $\varphi_\xi$ to the product of inverse-transformed child polynomials.

Each operation on degree-$n$ polynomials requires $O(n\log n)$ time (via FFT), and the cotree has $O(n)$ nodes, giving an overall $O(n^2\log n)$ complexity (Jumadildayev, 17 Dec 2025).

7. Analytic and Combinatorial Properties

By the correspondence with independence polynomials, one transfers real-rootedness, monotonicity, and log-concavity properties. For connected $G$, $h(G,x)$ has a unique largest positive real zero $\gamma(G)$, with $|\xi| < \gamma(G)$ for all other zeros. This zero satisfies $\gamma(G) \leq t(G) \leq 4(\Delta(G)-1)$ (where $t(G)$ is the largest zero of the matching polynomial). The sequence $\{(-1)^k a_{n-k}(G)\}$ is ultra-log-concave and unimodal. For every subgraph $H \subseteq G$ of connected $G$, the ratio $h^*(H,x)/h^*(G,x)$ (and correspondingly $I(H,x)/I(G,x)$) has non-negative integer coefficients, establishing coefficient-positivity and strict monotonicity under subgraph inclusion (Bencs, 2017).

Illustrative Table: Key Computational Forms

Object	Formula	Comments
Clique-cover polynomial	$\xi_G(t) = \sum_k s_k(G)\,t^k$	$s_k(G)$: number of $k$-clique covers
Duality (chromatic dual)	$\xi_G(t) = \varphi_\xi[\chi_{\overline G}(t)]$	$\varphi_\xi$: Stirling basis operator
Adjoint polynomial	$h(G,x) = \sum_{k=1}^n (-1)^{n-k} a_k(G)x^k$	$a_k(G)$: as above
Adjoint–Independence link	$h^*(G,x) = x^n h(G,1/x) = I(\widehat G,x)$	$\widehat G$: auxiliary graph

The structural interplay between clique covers, adjoint polynomials, Stirling transforms, chromatic polynomials, and independence polynomials provides a unified analytic and combinatorial framework for studying graph coverings and their algebraic properties (Jumadildayev, 17 Dec 2025, Bencs, 2017).