## 1 Introduction

A lattice polytope $${\mathscr {P}}\subset {\mathbb R}^n$$ is a convex polytope all of whose vertices have integer coordinates. Its normalized volume, $${{\,\mathrm{Vol}\,}}({\mathscr {P}})=\dim ({\mathscr {P}})! {{\,\mathrm{vol}\,}}({\mathscr {P}})$$ where $${{\,\mathrm{vol}\,}}({\mathscr {P}})$$ is the relative volume of $${\mathscr {P}}$$, is always a positive integer. To compute $${{\,\mathrm{Vol}\,}}({\mathscr {P}})$$ is a fundamental but hard problem in polyhedral geometry.

Let G be a simple graph on $$[n]:=\{1,\ldots ,n\}$$ with edge set E(G). The PV-type adjacency polytope $$\nabla ^\mathrm{PV}_G$$ of G is the lattice polytope which is the convex hull of

\begin{aligned}\{ \pm ({\mathbf{e}}_i-{\mathbf{e}}_j) \in {\mathbb R}^n : \{i,j\} \in E(G)\},\end{aligned}

where $${\mathbf{e}}_i$$ is the i-th unit coordinate vector in $${\mathbb R}^n$$. The normalized volumes of PV-type adjacency polytopes have attracted much attention. In fact, the normalized volume of a PV-type adjacency polytope gives an upper bound on the number of possible solutions in the Kuramoto equations [5], which models the behavior of interacting oscillators [13]. For several classes of graphs, explicit formulas for the normalized volume of their PV-type adjacency polytopes have been given (e.g., [1, 7, 10]). In particular, we can compute the normalized volume of the PV-type adjacency polytope of a suspension graph by using interior polynomials [15]. Here interior polynomials are a version of the Tutte polynomials for hypergraphs introduced by Kálmán [11].

On the other hand, the PQ-type adjacency polytope $$\nabla ^PQ _G$$ of G is the lattice polytope which is the convex hull of

\begin{aligned}\{ ({\mathbf{e}}_i,{\mathbf{e}}_j) \in {\mathbb R}^{2n} : \{i,j\} \in E(G)\; \text{ or }\; i=j\}.\end{aligned}

Note that an edge $$\{i,j\} \in E(G)$$ results in both $$({\mathbf{e}}_i, {\mathbf{e}}_j)$$ and $$({\mathbf{e}}_j, {\mathbf{e}}_i)$$. There is a connection between PQ-type adjacency polytopes and the engineering problem known as power-flow study, which models the balances of electric power on a network of power generation [6]. In fact, the normalized volume of a PQ-type adjacency polytope gives an upper bound on the number of possible solutions in the algebraic power-flow equations. In the present paper, we focus on the $$h^*$$-polynomial of a PQ-type adjacency polytope. Here, the $$h^*$$-polynomial $$h^*({\mathscr {P}},x)$$ of a lattice polytope $${\mathscr {P}}$$ is a discrete tool to compute the normalized volume $${{\,\mathrm{Vol}\,}}({\mathscr {P}})$$ (see Sect. 2).

We recall a relation between $$\nabla ^PQ _G$$ and a root polytope. For a bipartite graph H on [n] with edge set E(H), the root polytope $${\mathcal Q}_H$$ of H is the lattice polytope which is the convex hull of

\begin{aligned}\{{\mathbf{e}}_i+ {\mathbf{e}}_j \in {\mathbb R}^n:\{i,j\} \in E(H)\}.\end{aligned}

For a positive integer n, set $$[\overline{n}]:=\{\overline{1},\ldots ,\overline{n}\}$$. Define D(G) to be the bipartite graph on $$[n] \cup [\overline{n}]$$ with edges $$\{i, \overline{i}\}$$ for each $$i \in [n]$$ and $$\{i, \overline{j}\}$$ and $$\{\overline{i},j\}$$ for each edge $$\{i,j\}$$ in G. It then follows that $$\nabla ^PQ _G$$ is unimodularly equivalent to $${\mathcal Q}_{D(G)}$$ [8, Lem. 2.4]. On the other hand, it is known [12] that the $$h^*$$-polynomial of the root polytope $${\mathcal Q}_H$$ of a connected bipartite graph H coincides with the interior polynomial $$I_H(x)$$ of the associated hypergraph of H. In particular, the normalized volume of $${\mathcal Q}_H$$ is equal to $$|{{{\,\mathrm{HT}\,}}(H)}|$$, where $${{\,\mathrm{HT}\,}}(H)$$ denotes the set of hypertrees of an associated hypergraph of H. Therefore, we can compute the $$h^*$$-polynomial and the normalized volume of $$\nabla ^PQ _G$$ of a connected graph G by using an interior polynomial and counting hypertrees. In the terminology of [17], a hypertree is called a draconian sequence [17, Defn. 9.2]. Moreover, Davis and Chen [8] have studied the normalized volume of $$\nabla ^PQ _G$$ by using draconian sequences.

The main results of the present paper are formulas of the $$h^*$$-polynomial and the normalized volume of $$\nabla ^PQ _G$$ of a join graph G. Let $$G_1,\ldots ,G_s$$ be graphs with $$m_1,\ldots ,m_s$$ vertices. Suppose that $$G_i$$ and $$G_j$$ have no common vertices for each $$i\ne j$$. Then the join $$G_1 + \cdots + G_s$$ of $$G_1,\ldots ,G_s$$ is obtained from $$G_1 \cup \cdots \cup G_s$$ joining each vertex of $$G_i$$ to each vertex of $$G_j$$ for any $$i \ne j$$. Note that $$G_1 + \cdots + G_s$$ $$(s>1)$$ is connected and hence so is $$D(G_1 + \cdots + G_s)$$. For example, the complete bipartite graph $$K_{\ell ,m}$$ is equal to the join $$E_\ell + E_m$$ where $$E_k$$ is the empty graph with k vertices. For complete graphs $$K_\ell$$ and $$K_m$$, one has $$K_\ell + K_m = K_{\ell +m}$$. We can compute the $$h^*$$-polynomial and the normalized volume of the PQ-type adjacency polytope of a join graph by using perfectly matchable set polynomials (see Sect. 2 for the definition of perfectly matchable set polynomials).

### Theorem 1.1

Let $$G_1,\ldots ,G_s$$ be graphs with $$m_1,\ldots ,m_s$$ vertices, respectively. Suppose that $$G_i$$ and $$G_j$$ have no common vertices for each $$i \ne j$$. Then for the join $$G= G_1 + \cdots + G_s$$ with $$m = \sum _{i=1}^s m_i$$ vertices, we have

\begin{aligned} \begin{aligned} h^*\bigl (\nabla ^PQ _G ,x\bigr )&=\sum _{i=1}^s h^*\bigl (\nabla ^PQ _{G_i+K_{m-m_i}},x\bigr ) -(s-1)\sum _{k=0}^{m -1}\left( {\begin{array}{c}m-1\\ k\end{array}}\right) ^2x^k\\&=\sum _{i=1}^s I_{D(G_i+K_{m-m_i})}(x) -(s-1)\sum _{k=0}^{m -1}\left( {\begin{array}{c}m-1\\ k\end{array}}\right) ^2x^k\\&=\sum _{i=1}^sp(D(G_i+K_{m-m_i-1}), x)-(s-1)\sum _{k=0}^{m -1}\left( {\begin{array}{c}m-1\\ k\end{array}}\right) ^2x^k, \end{aligned} \end{aligned}
(1)

where $$p(H,x)$$ denotes the perfectly matchable set polynomial of a graph H. In particular, one has

\begin{aligned} {{\,\mathrm{Vol}\,}}\bigl (\nabla ^PQ _G\bigr )&=\sum _{i=1}^s{{\,\mathrm{Vol}\,}}\bigl (\nabla ^PQ _{G_i+K_{m-m_i}}\bigr )-(s-1)\left( {\begin{array}{c}2(m-1)\\ m-1\end{array}}\right) \\&=\sum _{i=1}^s|{{{\,\mathrm{HT}\,}}(D(G_i+K_{m-m_i}))}|-(s-1)\left( {\begin{array}{c}2(m-1)\\ m-1\end{array}}\right) \\&=\sum _{i=1}^s|PM (D(G_i+K_{m-m_i-1}))|-(s-1)\left( {\begin{array}{c}2(m-1)\\ m-1\end{array}}\right) , \end{aligned}

where $$PM (H)$$ denotes the set of perfectly matchable sets of a graph H.

By using this theorem, we give explicit formulas of the $$h^*$$-polynomial and the normalized volume of $$\nabla ^PQ _G$$ when G is a complete multipartite graph (Corollary 4.4). On the other hand, Theorem 1.1 is not useful for computing the $$h^*$$-polynomial and the normalized volume of $$\nabla ^PQ _{G}$$ when G is a wheel graph $$W_n$$, that is, G is the join of a cycle $$C_n$$ and $$K_1$$. We give explicit formulas for the $$h^*$$-polynomial and the normalized volume of $$\nabla _{W_n}^PQ$$ and prove the conjecture [8, Conj. 4.4] on the normalized volume of $$\nabla _{W_n}^PQ$$ (Theorem 5.1) by computing the perfectly matchable set polynomial of $$D(C_n)$$.

The paper is organized as follows: After reviewing the definitions and properties of the $$h^*$$-polynomials of lattice polytopes and the interior polynomials of connected bipartite graphs in Sect. 2, we give a proof of Theorem 1.1 in Sect. 3. By using Theorem 1.1, explicit formulas of the $$h^*$$-polynomial and the normalized volume of the PQ-type adjacency polytope of a complete multipartite graph are presented in Sect. 4. Finally, we compute the $$h^*$$-polynomial and the normalized volume of the PQ-type adjacency polytope of a wheel graph in Sect. 5.

## 2 Preliminaries

As explained in the previous section, the $$h^*$$-polynomial of $$\nabla ^PQ _G$$ is equal to the interior polynomial of D(G). First, we give a brief introduction of Ehrhart polynomials and $$h^*$$-polynomials. We refer the reader to [4] for the detailed information for them. Let $${\mathscr {P}}\subset {\mathbb R}^n$$ be a lattice polytope of dimension d. Given a positive integer t, we define

\begin{aligned}L_{{\mathscr {P}}}(t)=|t {\mathscr {P}}\cap {\mathbb Z}^n|,\end{aligned}

where $$t{\mathscr {P}}:=\{ t {\mathbf{x}}\in {\mathbb R}^n : {\mathbf{x}}\in {\mathscr {P}}\}$$. The study on $$L_{{\mathscr {P}}}(t)$$ originated in Ehrhart [9] who proved that $$L_{{\mathscr {P}}}(t)$$ is a polynomial in t of degree d with the constant term 1. We call $$L_{{\mathscr {P}}}(t)$$ the Ehrhart polynomial of $${\mathscr {P}}$$. The generating function of the lattice point enumerator, i.e., the formal power series

\begin{aligned}Ehr _{\mathscr {P}}(x)=1+\sum _{k=1}^{\infty }L_{{\mathscr {P}}}(k) x^k\end{aligned}

is called the Ehrhart series of $${\mathscr {P}}$$. It is known that it can be expressed as a rational function of the form

\begin{aligned}Ehr _{\mathscr {P}}(x)=\frac{h^*({\mathscr {P}},x)}{(1-x)^{d+1}},\end{aligned}

where $$h^*({\mathscr {P}},x)$$ is a polynomial in x of degree at most d with nonnegative integer coefficients called the $$h^*$$-polynomial (or the $$\delta$$-polynomial) of $${\mathscr {P}}$$. Moreover,

\begin{aligned}h^*({\mathscr {P}},x)=\sum _{i=0}^{d} h_i^* x^i\end{aligned}

satisfies $$h^*_0=1$$, $$h^*_1=|{\mathscr {P}}\cap {\mathbb Z}^n|-(d +1)$$, and $$h^*_{d}=|int ({\mathscr {P}}) \cap {\mathbb Z}^n|$$, where $$int ({\mathscr {P}})$$ is the relative interior of $${\mathscr {P}}$$. Furthermore, $$h^*({\mathscr {P}},1)=\sum _{i=0}^{d} h_i^*$$ is equal to the normalized volume $${{\,\mathrm{Vol}\,}}({\mathscr {P}})$$ of $${\mathscr {P}}$$.

Next, we recall the definition of interior polynomials and their properties. A hypergraph is a pair $${\mathcal H}= (V, E)$$, where $$E=\{e_1,\ldots ,e_n\}$$ is a finite multiset of non-empty subsets of $$V=\{v_1,\ldots ,v_m\}$$. Elements of V are called vertices and the elements of E are the hyperedges. Then we can associate $${\mathcal H}$$ with a bipartite graph $${{\,\mathrm{Bip}\,}}{\mathcal H}$$ on the vertex set $$E \sqcup V$$ with the edge set $$\{ \{e_j, v_i\} : v_i \in e_j\}$$. Assume that $${{\,\mathrm{Bip}\,}}{\mathcal H}$$ is connected. A hypertree in $${\mathcal H}$$ is a function $$f:E \rightarrow {\mathbb Z}_{\ge 0}$$ such that there exists a spanning tree $$\Gamma$$ of $${{\,\mathrm{Bip}\,}}{\mathcal H}$$ whose vertices have degree $$f (e) +1$$ at each $$e \in E$$. Then we say that $$\Gamma$$ induces f. For example, if $${\mathcal H}$$ is a hypergraph with $$V=\{v_1,v_2,v_3\}$$ and $$E=\{e_1=\{v_1,v_2,v_4\}, e_2=\{v_2,v_3,v_4\}\}$$, then $${{\,\mathrm{Bip}\,}}{\mathcal H}$$ is a bipartite graph whose edge set is $$\{ \{e_1, v_1\}, \{e_1, v_2\}, \{e_1, v_4\}, \{e_2, v_2\}, \{e_2, v_3\}, \{e_2, v_4\}\}$$. A spanning tree

\begin{aligned}\Gamma = \{ \{e_1, v_1\}, \{e_1, v_2\}, \{e_1, v_4\}, \{e_2, v_2\}, \{e_2, v_3\}\}\end{aligned}

of $${{\,\mathrm{Bip}\,}}{\mathcal H}$$ induces a hypertree $$f:E \rightarrow {\mathbb Z}_{\ge 0}$$ with $$f(e_1) = 2$$ and $$f(e_2) = 1$$. Let $${{\,\mathrm{HT}\,}}({\mathcal H})$$ denote the set of all hypertrees in $${\mathcal H}$$. A hyperedge $$e_j \in E$$ is said to be internally inactive with respect to the hypertree f if there exists $$j' < j$$ such that $$g:E \rightarrow {\mathbb Z}_{\ge 0}$$ defined by

\begin{aligned}g(e_i) = {\left\{ \begin{array}{ll}f(e_i)+1 &{}\quad \text {if}\; i=j',\\ f(e_i)-1 &{}\quad \text {if}\;i=j,\\ f(e_i) &{} \quad \text {otherwise},\end{array}\right. }\end{aligned}

is a hypertree. Note that it depends on the ordering $$e_1,\ldots ,e_n$$ of hyperedges. In particular, $$e_1$$ is not internally inactive in general. For example, with respect to the hypertree f with $$f(e_1) = 2$$ and $$f(e_2) = 1$$ above, there exists no internally inactive hyperedges. Let $$\overline{\iota } (f)$$ be the number of internally inactive hyperedges of f. Then the interior polynomial of $${\mathcal H}$$ is the generating function $$I_{\mathcal H}(x) = \sum _{f \in {{\,\mathrm{HT}\,}}({\mathcal H})} x^{ \overline{\iota } (f)}$$. It is known [11, Prop. 6.1] that $$\deg I_{\mathcal H}(x)\le \min {\{|V|,|E|\}}-1$$. If $$G ={{\,\mathrm{Bip}\,}}{\mathcal H}$$, then we set $${{\,\mathrm{HT}\,}}(G) ={{\,\mathrm{HT}\,}}({\mathcal H})$$ and $$I_G (x) = I_{\mathcal H}(x)$$. The coefficients of $$I_G(x)$$ are described as follows.

### Proposition 2.1

[11, Thm. 3.4] Let G be a connected bipartite graph on the vertex set $$V_1 \sqcup V_2$$ where $$V_1=\{v_1,\ldots ,v_p\}$$ and $$|V_2|=q$$. Then the coefficient of $$x^k$$ in $$I_G(x)$$ is the number of functions $$f:V_1 \rightarrow {\mathbb Z}_{\ge 0}$$ such that

1. (i)

$$\displaystyle \sum _{i=1}^p f(v_i) = q -1$$;

2. (ii)

$$\displaystyle \sum _{v \in V'} f(v) \le |\Gamma _G(V')| -1$$ for all $$V' \subset V_1$$, where $$\Gamma _G(S) \subset V_2$$ is the set of vertices adjacent to some vertex in S;

3. (iii)

$$\overline{\iota } (f) =k$$, i.e., $$|\eta _G(f)|=k$$ where $$\eta _G(f)$$ is the set of vertices $$v_j \in V_1$$ satisfying the following condition: there exists $$j' < j$$ such that the function $$g:V_1 \rightarrow {\mathbb Z}_{\ge 0}$$ defined by

\begin{aligned}g(v_i) = {\left\{ \begin{array}{ll}f(v_i)+1 &{}\quad \text {if}\; i=j',\\ f(v_i)-1 &{} \quad \text {if}\;i=j,\\ f(v_i) &{}\quad \text {otherwise},\end{array}\right. }\end{aligned}

satisfies condition (ii) above.

From [8, Lem. 2.4] and [12, Thms. 1.1 and 3.10], we have the following.

### Proposition 2.2

Let G be a connected graph. Then $$h^*(\nabla _G^PQ ,x) = I_{D(G)}(x)$$. In particular, the normalized volume of $$\nabla _G^PQ$$ is $$I_{D(G)}(1)=|{{{\,\mathrm{HT}\,}}(D(G))}|$$.

Let G be a finite graph on the vertex set $$V=[n]$$. A k-matching of G is a set of k pairwise non-adjacent edges of G. The matching generating polynomial of G is

\begin{aligned}g(G,x) =\sum _{k \ge 0} m_G(k) x^k,\end{aligned}

where $$m_G(k)$$ is the number of k-matchings of G. A k-matching of G is said to be perfect if $$2k =n$$. A subset $$S \subset V$$ is called a perfectly matchable set [2] if the induced subgraph of G on the vertex set S has a perfect matching. Let $$PM (G, k)$$ be the set of all perfectly matchable sets S of G with $$|S|=2k$$ and $$PM (G)$$ the set of all perfectly matchable sets of G. We regard $$\emptyset$$ as a perfectly matchable set and we set $$PM (G,0) = \{\emptyset \}$$. Note that $$|PM (G, k)| \le m_G(k)$$ holds in general. We call the polynomial

\begin{aligned}p(G,x) = \sum _{k \ge 0}|PM (G,k)| x^k\end{aligned}

the perfectly matchable set polynomial (PMS polynomial) of G.

### Example 2.3

For the cycle $$C_4$$ of length 4, we have $$g(C_4,x) = 2 x^2 + 4x+1$$ and $$p(C_4,x) = x^2 + 4x+1$$. On the other hand, if a graph G has no even cycles, then we have $$g(G,x) = p(G,x)$$.

Assume that G is a bipartite graph with a bipartition $$[n] =V_1 \sqcup V_2$$. Then let $$\widetilde{G}$$ be a connected bipartite graph on $$[n+2]$$ whose edge set is

\begin{aligned}E(\widetilde{G}) = E(G) \cup \{ \{i, n+1\} : i \in V_1\} \cup \{ \{j, n+2\} : j \in V_2 \cup \{n+1\}\}.\end{aligned}

### Proposition 2.4

[14, Prop. 3.4] Let G be a bipartite graph. Then we have $$I_{\widetilde{G}}(x)= p(G,x)$$.

Although the following lemma is easy to see, it will be useful.

### Lemma 2.5

Let G be a graph. Then we have $$D(G+K_1) = \widetilde{D(G)}$$.

Moreover, we have

### Proposition 2.6

Let G be a graph with n vertices. Then the $$h^*$$-polynomial of $$\nabla _{G+K_1}^PQ$$ is $$p(D(G), x)$$. In particular, the normalized volume of $$\nabla _{G+K_1}^PQ$$ is $$|PM (D(G))|$$.

### Proof

The assertion follows from Propositions 2.2, 2.4, and Lemma 2.5. $$\square$$

### Remark 2.7

Given a graph G, the number of matchings of G is called the Hosoya index of G and is denoted by Z(G). From Proposition 2.6, the normalized volume of $$\nabla _{G+K_1}^PQ$$ is at most Z(D(G)).

Let G be a graph on the vertex set [n]. Given a subset $$S \subset [n]$$, we associate the (0, 1)-vector $$\rho (S)=\sum _{i \in S} \mathbf{e}_i \in {\mathbb R}^n$$. For example, $$\rho (\emptyset ) = \mathbf{0} \in {\mathbb R}^n$$. The convex hull of

\begin{aligned}\{ \rho (S) : S \in PM (G,k) \text{ for } \text{ some }\; k \}\end{aligned}

is called a perfectly matchable subgraph polytope (PMS polytope) of G. A system of linear inequalities for a PMS polytope of G was given in [2] for bipartite graphs, and in [3] for arbitrary graphs.

### Proposition 2.8

[2, Thm. 1] Let G be a bipartite graph on the vertex set $$[n]=V_1 \sqcup V_2$$. Then the PMS polytope of G is the set of all vectors $$(x_1,\ldots ,x_n) \in {\mathbb R}^n$$ such that

\begin{aligned} 0&\le x_i \le 1\quad \text {for each}\quad i=1,2,\ldots ,n,\\ \sum _{i \in V_1} x_i&= \sum _{j \in V_2} x_j,\\ \sum _{i \in S} x_i&\le \sum _{j \in \Gamma _G(S)}\!\! x_j \quad \text{ for } \text{ all }\quad S \subset V_1, \end{aligned}

where $$\Gamma _G(S) \subset V_2$$ is the set of vertices adjacent to some vertex in S.

## 3 Join Graphs

In the present section, we give a proof of Theorem 1.1. Given a graph G and a nonnegative integer k, let $$F_G(k)$$ be the set of functions satisfying conditions (i)–(iii) in Proposition 2.1 for D(G).

### Lemma 3.1

Let $$G_1$$ and $$G_2$$ be graphs with $$m_1$$ and $$m_2$$ vertices, respectively. Suppose that $$G_1$$ and $$G_2$$ have no common vertices. Then for the join $$G= G_1 +G_2$$ with $$m = m_1+m_2$$ vertices, we have

\begin{aligned}&\bigsqcup _{k} F_G(k) = \bigsqcup _{k}\;F_{G_1+K_{m_2}}(k)\,\cap \,\bigsqcup _{k}\;F_{K_{m_1}+G_2}(k), \end{aligned}
(2)
\begin{aligned}&\bigsqcup _{k}\;F_{G_1+K_{m_2}}(k) \cup \bigsqcup _{k} \, F_{K_{m_1}+G_2}(k)=\bigsqcup _{k} F_{K_m}(k). \end{aligned}
(3)

### Proof

Let $$V_1 \sqcup V_2$$ with $$V_1=\{v_1,\ldots , v_{m}\}$$ and $$V_2=\{v_1',\ldots , v_{m}'\}$$ denote the common vertex set of D(G), $$D(G_1+K_{m_2})$$, $$D(K_{m_1}+G_2)$$, and $$D(K_m)$$ $$(=K_{m,m})$$. In addition, let $$V_{1,1} =\{v_1,\ldots , v_{m_1}\}$$, $$V_{1,2} =\{v_{m_1+1},\ldots , v_m\}$$, $$V_{2,1} =\{v_1',\ldots , v_{m_1}'\}$$, and $$V_{2,2} =\{v_{m_1+1}',\ldots , v_m'\}$$, where $$V_{1,j} \sqcup V_{2,j}$$ corresponds to the vertex set of $$D(G_j)$$ for $$j =1,2$$. First we will show that (2) holds. Let H be one of D(G), $$D(G_1+K_{m_2})$$, $$D(K_{m_1}+G_2)$$. Since $$\bigsqcup _{k} F_H(k)$$ consists of functions satisfying conditions (i) and (ii) in Proposition 2.1, we study relations between conditions (i) and (ii) as follows:

Case 1  ($$H = D(G_1+K_{m_2})$$). Suppose that f satisfies condition (i). If $$v \in V' \subset V_1$$ for some $$v \in V_{1,2}$$, then condition (ii) holds for $$V'$$ since $$\deg (v) = m$$. Thus, if f satisfies condition (i), then condition (ii) in Proposition 2.1 is equivalent to the condition

\begin{aligned} \sum _{v \in V'} f(v) \le |\Gamma _{D(G_1)}(V')| + m_2 -1\quad \text{ for } \text{ all }\quad V' \subset V_{1,1}. \end{aligned}
(4)

Case 2  ($$H = D(K_{m_1}+G_2)$$). By the similar argument as in Case 1, if f satisfies condition (i), then condition (ii) in Proposition 2.1 is equivalent to the condition

\begin{aligned} \sum _{v \in V'} f(v) \le |\Gamma _{D(G_2)}(V')| + m_1 -1\quad \text{ for } \text{ all }\quad V' \subset V_{1,2}. \end{aligned}
(5)

Case 3  ($$H = D(G)$$). Suppose that f satisfies condition (i). If $$v,v' \in V' \subset V_1$$ for some $$v \in V_{1,1}$$ and $$v' \in V_{1,2}$$, then condition (ii) holds for $$V'$$ since $$\Gamma _{D(G)}(\{v,v'\}) =V_2$$. Hence, if f satisfies (i), then condition (ii) in Proposition 2.1 holds if and only if both (4) and (5) hold.

Hence (2) holds.

For the graph $$D(K_m)$$, since $$|\Gamma _{D(K_m)}(V')| -1 = m-1$$ for all $$V' \subset V_1$$, condition (i) implies condition (ii) in Proposition 2.1. Since any element in $$F_{G_1+K_{m_2}}(k) \cup F_{K_{m_1}+G_2}(k)$$ satisfies condition (i), we have

\begin{aligned}\bigsqcup _{k}\,F_{G_1+K_{m_2}}(k) \,\cup \,\bigsqcup _{k}\,F_{K_{m_1}+G_2}(k)\,\subset \,\bigsqcup _{k}\, F_{K_m}(k).\end{aligned}

Suppose that f satisfies none of (4) and (5). Then

\begin{aligned} \sum _{v \in V'} f(v)&\ge |\Gamma _{D(G_1)}(V')| + m_2\quad \text {for some}\quad V' \subset V_{1,1},\\ \sum _{v \in V''} f(v)&\ge |\Gamma _{D(G_2)}(V'')| + m_1\quad \text {for some}\quad V'' \subset V_{1,2}. \end{aligned}

It then follows that $$\sum _{v \in V_1} f(v) \ge m_1 + m_2 =m$$. Hence, if f satisfies condition (i), then at least one of (4) or (5) holds. Thus (3) holds. $$\square$$

### Lemma 3.2

Let $$G_1$$ and $$G_2$$ be graphs with $$m_1$$ and $$m_2$$ vertices, respectively. Suppose that $$G_1$$ and $$G_2$$ have no common vertices. Then for the join $$G= G_1 +G_2$$ with $$m = m_1+m_2$$ vertices, $$F_{K_m}(k)$$ is decomposed into the disjoint sets

\begin{aligned}F_{K_m}(k)=F_{K_{m_1}+G_2}(k) \sqcup (F_{G_1+K_{m_2}}(k) \setminus F_G(k))\end{aligned}

as in Fig. 1.

### Proof

Let $$V_1 \sqcup V_2$$ with $$V_1=\{v_1,\ldots , v_{m}\}$$ and $$V_2=\{v_1',\ldots , v_{m}'\}$$ denote the common vertex set of D(G), $$D(G_1+K_{m_2})$$, $$D(K_{m_1}+G_2)$$, and $$D(K_m)$$ $$(=K_{m,m})$$. In addition, let $$V_{1,1} =\{v_1,\ldots , v_{m_1}\}$$, $$V_{1,2} =\{v_{m_1+1},\ldots , v_m\}$$, $$V_{2,1} =\{v_1',\ldots , v_{m_1}'\}$$, and $$V_{2,2} =\{v_{m_1+1}',\ldots , v_m'\}$$, where $$V_{1,j} \sqcup V_{2,j}$$ corresponds to the vertex set of $$D(G_j)$$ for $$j =1,2$$.

### Claim 1

$$F_{K_{m_1}+G_2}(k) \subset F_{K_m}(k)$$.

Suppose that f belongs to $$F_{K_{m_1}+G_2}(k)$$. From Lemma 3.1 (3), we have $$f \in F_{K_m}(\ell )$$ for some $$\ell$$. Then condition (ii) in Proposition 2.1 is independent from the value $$f(v_1)$$ since $$\deg (v_1) =m$$. Thus one can choose $$j'=1$$ for condition (iii) for any $$j >1$$, and hence we have

\begin{aligned}k = |\{v_j \in V_1 : v_j \ne v_1, \,f(v_j) >0\}| = \ell .\end{aligned}

Therefore

\begin{aligned} F_{K_{m_1}+G_2}(k) \subset F_{K_m}(k) \end{aligned}
(6)

and $$F_{K_{m_1}+G_2}(k) \cap F_{K_m}(\ell ) =\emptyset$$ if $$k \ne \ell$$.

### Claim 2

$$F_G(k) \subset F_{G_1+K_{m_2}}(k)$$.

Suppose that $$f \in F_G(k)$$. From (2) of Lemma 3.1 we have $$f \in F_{G_1+K_{m_2}}(\ell )$$ for some $$\ell$$. Note that $$k = |\eta _{D(G)} (f)|$$ and $$\ell = |\eta _{D(G_1+K_{m_2})} (f)|$$ (where $$\eta$$ is defined in Proposition 2.1). In order to prove $$k=\ell$$, we will show that $$\eta _{D(G)} (f) = \eta _{D(G_1+K_{m_2})} (f)$$. Suppose that $$f(v_j) >0$$ for $$v_j \in V_1$$, $$j \ne 1$$.

Case 1  ($$j \le m_1+1$$). From the argument in proof of Lemma 3.1, for the graph $$D(G_1+K_{m_2})$$ (resp. D(G)), g defined in condition (iii) of Proposition 2.1 satisfies condition (ii) if and only if g satisfies (4) (resp. both (4) and (5)). Moreover, since the vertex $$v_{j'}$$ in condition (iii) in Proposition 2.1 should be chosen from $$V_{1,1}$$, it follows that g satisfies (5) if f satisfies (5). Thus $$v_j$$ belongs to $$\eta _{D(G)}(f)$$ if and only if $$v_j$$ belongs to $$\eta _{D(G_1+K_{m_2})} (f)$$.

Case 2  ($$j \ge m_1+2$$). Since $$v_{m_1+1}$$ belongs to $$V_{1,2}$$ with $$m_1+1 < j$$, (4) holds for g defined in condition (iii) of Proposition 2.1 where $$v_{j'}=v_{m_1+1}$$. Thus $$v_{j'}=v_{m_1+1}$$ satisfies condition (iii) of Proposition 2.1 for $$v_j$$ in $$D(G_1+K_{m_2})$$. Hence $$v_j$$ belongs to $$\eta _{D(G_1+K_{m_2})}(f)$$. We now show that any $$v_j$$, $$j \ge m_1+2$$, belongs to $$\eta _{D(G)} (f)$$, i.e., there exists $$j' < j$$ such that $$g:V_1 \rightarrow {\mathbb Z}_{\ge 0}$$ defined by

\begin{aligned} g(v_i) = {\left\{ \begin{array}{ll}f(v_i)+1 &{}\quad \text {if}\;i=j', \\ f(v_i)-1 &{}\quad \text {if}\;i=j,\\ f(v_i) &{} \quad \text {otherwise},\end{array}\right. } \end{aligned}
(7)

is a hypertree in D(G). Let $$\Gamma$$ be a spanning tree of D(G) that induces f. Suppose that $$\Gamma$$ does not contain an edge $$e=\{v_k, v_1'\}$$ for some $$m_1< k \le m$$. Then $$\Gamma \cup \{e\}$$ has a unique cycle, and the cycle contains $$v_k$$. Let $$e'$$ be the edge of the cycle that is adjacent to $$v_k$$ but different from e. Then the spanning tree $$(\Gamma \cup \{e\}) \setminus \{e'\}$$ induces f since the degree of each $$v_i \in V_1$$ is same for $$\Gamma$$ and $$(\Gamma \cup \{e\}) \setminus \{e'\}$$. Thus we may assume that $$\{ \{v_k, v_1'\} : m_1 <k\le m \} \subset \Gamma$$. By the same argument, we may assume that $$\{ \{v_k, v_{m_1+1}'\} : 1\le k \le m_1\}$$ is a subset of $$\Gamma$$. Hence

\begin{aligned}S=\{ \{v_1, v_{m_1+1}'\} ,\ldots , \{v_{m_1}, v_{m_1+1}'\} ,\{v_{m_1+1},v_1'\}, \ldots , \{v_{m},v_1'\} \}\end{aligned}

is a subset of $$\Gamma$$. Since $$\Gamma$$ is spanning,

\begin{aligned}T=\{\{v_{i_2}, v_2'\} , \ldots , \{v_{i_{m_1}}, v_{m_1}'\},\{v_{i_{m_1+2}}, v_{m_1+2}'\} , \ldots , \{v_{i_{m}}, v_m'\}\}\end{aligned}

is a subset of $$\Gamma$$ for some $$1 \le i_2, \ldots , i_{m_1}, i_{m_1+2}, \ldots ,i_m \le m$$. Then $$S\cap T = \emptyset$$ and $$|S \cup T| = 2m-2$$. Since $$\Gamma$$ has $$2m-1$$ edges, we have

\begin{aligned}\Gamma = S \sqcup T \sqcup \{e\}\end{aligned}

for some edge e of D(G). In particular, the degree of each $$v_i'$$, $$i \in [m] \setminus \{1,m_1+1\}$$, in $$\Gamma$$ is at most 2. Since $$f(v_j) >0$$, the degree of $$v_j$$ in $$\Gamma$$ is at least 2, and hence $$\{v_j,v_k'\}$$ is an edge of $$\Gamma$$ for some $$k \in [m]\setminus \{1\}$$. Let $$e'=\{v_j,v_k'\}$$. We now construct a spanning tree of D(G) that induces a hypertree of the form (7). Note that, for an edge $$e''$$ of D(G), $$\Gamma '=(\Gamma \cup \{e''\} ) \setminus \{e'\}$$ is a spanning tree of D(G) if $$e'' \notin \Gamma$$ and the (unique) cycle of $$\Gamma \cup \{e''\}$$ contains $$e'$$.

Case 2.A  ($$k =m_1+1$$). Then $$\Gamma$$ contains a path $$(v_1', v_j, v_{m_1+1}', v_1)$$. Since $$\Gamma$$ has no cycles, $$\{v_1,v_1'\}$$ is not an edge of $$\Gamma$$. Hence $$(\Gamma \cup \{v_1, v_1'\} ) \setminus \{e'\}$$ is a spanning tree of D(G) that induces a hypertree of the form (7) where $$j'=1$$.

Case 2.B  ($$k \ne m_1+1$$ and $$\deg (v_k') =1$$). If $$2 \le k \le m_1$$, then $$(\Gamma \cup \{v_k, v_k'\} ) \setminus \{e'\}$$ is a spanning tree of D(G) that induces a hypertree of the form (7) where $$j'=k$$. If $$m_1+2 \le k \le m$$, then $$(\Gamma \cup \{v_1, v_k'\} ) \setminus \{e'\}$$ is a spanning tree of D(G) that induces a hypertree of the form (7) where $$j'=1$$.

Case 2.C  ($$k \ne m_1+1$$ and $$\deg (v_k') =2$$). Then $$\Gamma$$ contains an edge $$\{v_\ell ,v_k'\}$$ for some $$\ell \ne j$$. If $$m_1 < \ell \le m$$, then $$\Gamma$$ contains a cycle $$(v_1', v_j, v_k', v_\ell , v_1')$$ of length 4. This is a contradiction. Hence $$\ell \le m_1$$. Then $$\Gamma$$ contains a path $$(v_1', v_j, v_k', v_\ell ,v_{m_1+1}', v_1)$$. Since $$\Gamma$$ has no cycles, $$\{v_1,v_1'\}$$ is not an edge of $$\Gamma$$. Hence $$(\Gamma \cup \{v_1, v_1'\} ) \setminus \{e'\}$$ is a spanning tree of D(G) that induces a hypertree of the form (7) where $$j'=1$$.

Thus we have $$k=\ell$$, and hence

\begin{aligned} F_G(k) \subset F_{G_1+K_{m_2}}(k). \end{aligned}
(8)

### Claim 3

$$(F_{G_1+K_{m_2}}(k) \setminus F_G(k)) \subset F_{K_m}(k)$$.

Suppose that f belongs to $$F_{G_1+K_{m_2}}(k) \setminus F_G(k)$$. Then f does not satisfy (5), that is, there exists $$V' \subset V_{1,2}$$ such that

\begin{aligned}\sum _{v \in V'} f(v) \ge |\Gamma _{D(G_2)}(V')| +m_1.\end{aligned}

In particular, $$\sum _{v \in V_{1,2}} f(v) > m_1$$. Since $$\sum _{v \in V_1} f(v) \!=\! m \!-\!1$$, $$\sum _{v \!\in V_{1,1}} f(v) \!<\! m_2 \!-\! 1$$. Hence

\begin{aligned}\sum _{v \in V''} f(v)< m_2 - 1 < |\Gamma _{D(G_1)}(V'')| + m_2 - 1\end{aligned}

for all $$\emptyset \ne V'' \subset V_{1,1}$$. Thus, for each $$v_j \ne v_1$$ with $$f(v_j) >0$$, $$g:V_1 \rightarrow {\mathbb Z}_{\ge 0}$$ defined by

\begin{aligned}g(v_i) = {\left\{ \begin{array}{ll}f(v_i)+1 &{} \quad \text {if}\; i=1,\\ f(v_i)-1 &{} \quad \text {if}\;i=j,\\ f(v_i) &{} \quad \text {otherwise},\end{array}\right. }\end{aligned}

satisfies

\begin{aligned}\sum _{v \in V''} g(v) \le 1+\sum _{v \in V''} f(v) \le |\Gamma _{D(G_1)}(V'')| + m_2 - 1\end{aligned}

for all $$\emptyset \ne V'' \subset V_{1,1}$$. Hence g satisfies (4) in the proof of Lemma 3.1. Therefore we have

\begin{aligned}k = |\{v_j \in V_1 : v_j \ne v_1, \,f(v_j) >0\}|,\end{aligned}

and hence

\begin{aligned} (F_{G_1+K_{m_2}}(k) \setminus F_G(k)) \subset F_{K_m}(k). \end{aligned}
(9)

### Claim 4

$$F_{K_m}(k)=F_{K_{m_1}+G_2}(k) \cup (F_{G_1+K_{m_2}}(k) \setminus F_G(k))$$.

From (6) and (9),

\begin{aligned}F_{K_m}(k)\supset F_{K_{m_1}+G_2}(k) \cup (F_{G_1+K_{m_2}}(k) \setminus F_G(k)).\end{aligned}

Let $$f \in F_{K_m}(k) \setminus F_{K_{m_1}+G_2}(k)$$. Since $$F_{K_{m_1}+G_2}(k') \cap F_{K_m}(k) =\emptyset$$ for any $$k'\ne k$$, it follows that $$f \notin F_{K_{m_1}+G_2}(\ell )$$ for any $$\ell$$. Hence from (3), $$f \in F_{G_1+K_{m_2}}(\ell )$$ for some $$\ell$$. If $$f \in F_G(\ell )$$, then $$f \in F_{K_{m_1}+G_2}(\ell ')$$ for some $$\ell '$$ by (2). This is a contradiction. Thus $$f \in F_{G_1+K_{m_2}}(\ell ) \setminus F_G(\ell )$$. From (9), we have

\begin{aligned}f \in F_{G_1+K_{m_2}}(\ell ) \setminus F_G(\ell )\subset F_{K_m}(\ell ).\end{aligned}

Then $$\ell =k$$. It follows that

\begin{aligned}F_{K_m}(k)=F_{K_{m_1}+G_2}(k) \cup (F_{G_1+K_{m_2}}(k) \setminus F_G(k)).\end{aligned}

Finally, we show that this is a decomposition. Suppose that $$f \in F_{G_1+K_{m_2}}(k) \cap F_{K_{m_1}+G_2}(k)$$. From (2), $$f \in F_G(\ell )$$ for some $$\ell$$. Moreover, from (8), we have $$\ell = k$$. Thus

\begin{aligned} F_{G_1+K_{m_2}}(k) \cap F_{K_{m_1}+G_2}(k) \subset F_G(k).\end{aligned}

Therefore $$F_{K_m}(k)$$ is decomposed into the disjoint sets

\begin{aligned}F_{K_m}(k)=F_{K_{m_1}+G_2}(k) \sqcup (F_{G_1+K_{m_2}}(k) \setminus F_G(k))\end{aligned}

as in Fig. 1. $$\square$$

Now, we are in the position to give a proof of Theorem 1.1.

### Proof of Theorem 1.1

We prove this by induction on s. First we discuss the case when $$s=2$$, i.e., $$G= G_1+G_2$$. It is known [12, Exam. 5.3] that

\begin{aligned}|F_{K_m}(k)| = \left( {\begin{array}{c}m - 1\\ k\end{array}}\right) ^{\!2}.\end{aligned}

From Lemma 3.2,

\begin{aligned}\left( {\begin{array}{c}m - 1\\ k\end{array}}\right) ^{\!2} = |F_{K_{m_1}+G_2}(k)| + (|F_{G_1+K_{m_2}}(k) | - |F_G(k)| ).\end{aligned}

Thus we have

\begin{aligned}I_{D(G)}(x)=I_{D(G_1+K_{m_2})}(x) + I_{D(G_2+K_{m_1})}(x) -\sum _{k=0}^{m -1}\left( {\begin{array}{c}m-1\\ k\end{array}}\right) ^{\!2}x^k.\end{aligned}

Let $$s>2$$ and assume that the assertion holds for the join of at most $$s-1$$ graphs. Since $$G = (G_1 + \cdots + G_{s-1}) + G_s$$, we have

\begin{aligned} I_{D(G)}(x)&=I_{D((G_1 + \cdots + G_{s-1})+K_{m_s})}(x) + I_{D(G_s+K_{m - m_s})}(x) -\sum _{k=0}^{m -1}\left( {\begin{array}{c}m-1\\ k\end{array}}\right) ^2x^k\\&=\sum _{i=1}^{s-2} I_{D(G_i+K_{m-m_i})}(x)+I_{D((G_{s-1}+K_{m_s})+K_{m-m_{s-1}-m_s})}(x)\\&\qquad -(s-2)\sum _{k=0}^{m -1}\left( {\begin{array}{c}m-1\\ k\end{array}}\right) ^{\!2}x^k+ I_{D(G_s+K_{m - m_s})}(x) -\sum _{k=0}^{m -1}\left( {\begin{array}{c}m-1\\ k\end{array}}\right) ^2x^k\\&=\sum _{i=1}^s I_{D(G_i+K_{m-m_i})}(x) -(s-1)\sum _{k=0}^{m -1}\left( {\begin{array}{c}m-1\\ k\end{array}}\right) ^{\!2}x^k. \end{aligned}

From Proposition 2.4 and Lemma 2.5, this is equation (1). $$\square$$

## 4 Complete Multipartite Graphs

In this section, applying Theorem 1.1, we give explicit formulas for the $$h^*$$-polynomial and the normalized volume of the PQ-type adjacency polytope of a complete multipartite graph. Given positive integers $$\ell$$ and m, let

\begin{aligned}f_{\ell ,m}(x) = \sum _{k=0}^{\ell + m-1}\sum _{\alpha =0}^{k}\left( {\begin{array}{c}\ell -1\\ k-\alpha \end{array}}\right) \left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \sum _{\beta =\alpha }^{k}\left( {\begin{array}{c}\ell +\alpha -1\\ \beta \end{array}}\right) \left( {\begin{array}{c}m-\alpha \\ k-\beta \end{array}}\right) x^k.\end{aligned}

Since

\begin{aligned}&\sum _{k=0}^{\ell + m-1}\sum _{\alpha =0}^{k}\left( {\begin{array}{c}\ell -1\\ k-\alpha \end{array}}\right) \left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \sum _{\beta =0}^{k}\left( {\begin{array}{c}\ell +\alpha -1\\ \beta \end{array}}\right) \left( {\begin{array}{c}m-\alpha \\ k-\beta \end{array}}\right) x^k\\&\quad =\sum _{k=0}^{\ell + m-1}\sum _{\alpha =0}^{k}\left( {\begin{array}{c}\ell -1\\ k-\alpha \end{array}}\right) \left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \left( {\begin{array}{c}\ell +m-1\\ k\end{array}}\right) x^k\\&\quad =\sum _{k=0}^{\ell + m-1}\left( {\begin{array}{c}\ell +m-1\\ k\end{array}}\right) \sum _{\alpha =0}^{k}\left( {\begin{array}{c}\ell -1\\ k-\alpha \end{array}}\right) \left( {\begin{array}{c}m\\ \alpha \end{array}}\right) x^k=\sum _{k=0}^{\ell + m-1}\left( {\begin{array}{c}\ell +m-1\\ k\end{array}}\right) ^{\!2}x^k \end{aligned}

holds, we have

\begin{aligned}f_{\ell ,m}(x)= & {} \sum _{k=0}^{\ell +m -1}\left( {\begin{array}{c}\ell +m-1\\ k\end{array}}\right) ^{\!2} x^k-\sum _{k=1}^{m-1}\sum _{\alpha =1}^{k}\left( {\begin{array}{c}\ell -1\\ k-\alpha \end{array}}\right) \left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \\&\quad \times \sum _{\beta =0}^{\alpha -1}\left( {\begin{array}{c}\ell +\alpha -1\\ \beta \end{array}}\right) \left( {\begin{array}{c}m-\alpha \\ k-\beta \end{array}}\right) x^k.\end{aligned}

The $$h^*$$-polynomial of $$\nabla _G^PQ$$ for the graph $$G= K_\ell +E_m$$ $$(=K_{1,\ldots ,1,m})$$ coincides with $$f_{\ell ,m} (x)$$.

### Theorem 4.1

Let $$G= K_\ell +E_m$$. Then we have

\begin{aligned} h^*\bigl (\nabla _G^PQ ,x\bigr )&=f_{\ell ,m}(x),\\ {{\,\mathrm{Vol}\,}}\bigl (\nabla _G^PQ \bigr )&=f_{\ell ,m}(1)=\sum _{\alpha =0}^{m}\left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \sum _{\beta =0}^{\ell -1}\left( {\begin{array}{c}\ell +\alpha -1\\ \beta \end{array}}\right) \left( {\begin{array}{c}\ell +m-\alpha -1\\ \beta \end{array}}\right) . \end{aligned}

### Proof

Let $$G'=D(K_{\ell -1}+E_m)$$. Since $$G =(K_{\ell -1}+E_m) +K_1$$, from Proposition 2.6, we have

\begin{aligned}h^*\bigl (\nabla _G^PQ ,x\bigr )\,=\,p(G',x)\,=\!\sum _{k = 0}^{\ell + m-1}|{PM (G' ,k)}| x^k.\end{aligned}

Let $$n =\ell +m-1$$ and let $$[n] \cup [\overline{n}]$$ be the vertex set of $$G'$$. We decompose [n] into two disjoint sets $$V_{1,1} = [\ell -1]$$ and $$V_{1,2} = [n]\setminus [\ell -1]$$ where $$V_{1,1}$$ (resp. $$V_{1,2}$$) corresponds to $$K_{\ell -1}$$ (resp. $$E_m$$). Similarly, we decompose $$[\overline{n}]$$ into two disjoint sets $$V_{2,1}$$ and $$V_{2,2}$$. From Proposition 2.8, each $$|PM (G' ,k)|$$ is the number of (0, 1)-vectors $$(x_1,\ldots ,x_n, y_1,\ldots ,y_n) \in {\mathbb R}^{2n}$$ such that

\begin{aligned} \sum _{i = 1}^n x_i&=\sum _{i = 1}^n y_i = k, \end{aligned}
(10)
\begin{aligned} \sum _{i \in S} x_i&\le \sum _{\overline{j} \in \Gamma _{G'}(S)} y_j \quad \text {for all}\quad S \subset [n]. \end{aligned}
(11)

If (10) holds and a subset $$S \subset [n]$$ contains an element of $$V_{1,1}$$, then we have $$\Gamma _{G'}(S)=[\overline{n}]$$, and hence

\begin{aligned} \sum _{\overline{j} \in \Gamma _{G'}(S)} y_j=k=\sum _{i = 1}^n x_i \ge \sum _{i \in S} x_i.\end{aligned}

Thus (10) and (11) hold if and only if

\begin{aligned} \sum _{i = 1}^n x_i&=\sum _{i = 1}^n y_i = k, \end{aligned}
(12)
\begin{aligned} \sum _{i \in S} x_i&\le \sum _{\overline{j} \in \Gamma _{G'}(S)} y_j \quad \text {for all}\quad S \subset V_{1,2}. \end{aligned}
(13)

We count such vectors with $$|\{ i \in V_{1,2} : x_i = 1 \}|=\alpha$$ for each $$\alpha = 0,1,\ldots , k$$. Let $$S'=\{ i \in V_{1,2} : x_i = 1 \} \subset V_{1,2}$$.

Case 1  ($$\alpha >0$$). There are $$\left( {\begin{array}{c}m\\ \alpha \end{array}}\right)$$ possibilities for the choice of the subset $$S'$$. For each $$S'$$, there are $$\left( {\begin{array}{c}\ell -1\\ k-\alpha \end{array}}\right)$$ possibilities for the choice of the subset $$\{ i \in V_{1,1} : x_i = 1 \} \subset V_{1,1}$$. Then (12) and (13) hold if and only if $$\sum _{i = 1}^n y_i = k$$ and $$\alpha \le \sum _{\overline{j} \in \Gamma _{G'}(S')} y_j$$. Let $$\beta =\sum _{\overline{j} \in \Gamma _{G'}(S')} y_j$$. Since $$|\Gamma _{G'}(S')| = \ell + \alpha -1$$, there are $$\left( {\begin{array}{c}\ell +\alpha -1\\ \beta \end{array}}\right)$$ possibilities for the choice of the subset $$S''=\{ \overline{j} \in \Gamma _{G'}(S') : y_j = 1 \} \subset \Gamma _{G'}(S')$$. For each $$S''$$, there are $$\left( {\begin{array}{c}m-\alpha \\ k-\beta \end{array}}\right)$$ possibilities for the choice of the subset $$\{ \overline{j} \in [\overline{n}] \setminus \Gamma _{G'}(S') : y_j = 1 \}\subset [\overline{n}] \setminus \Gamma _{G'}(S')$$.

Case 2  ($$\alpha = 0$$). There are $$\left( {\begin{array}{c}\ell -1\\ k\end{array}}\right)$$ possibilities for the choice of the subset $$\{ i \in V_{1,1} : x_i = 1 \} \subset V_{1,1}$$. Let $$\beta = \sum _{\overline{j} \in V_{2,1}} y_j$$. Since $$S'=\emptyset$$, condition (13) always holds. Hence there are $$\left( {\begin{array}{c}\ell -1\\ \beta \end{array}}\right)$$ possibilities for the choice of the subset $$\{\overline{j} \in V_{2,1} : y_j = 1 \}$$, and there are $$\left( {\begin{array}{c}m\\ k-\beta \end{array}}\right)$$ possibilities for the choice of the subset $$\{\overline{j} \in V_{2,2} : y_j = 1 \}$$.

Thus we have $$I_{D(G)}(x)=f_{\ell ,m}(x)$$. Moreover, the normalized volume of $$\nabla _G^PQ$$ is equal to

\begin{aligned} f_{\ell ,m}(1)&=\sum _{k=0}^{\ell + m-1}\sum _{\alpha =0}^{k}\left( {\begin{array}{c}\ell -1\\ k-\alpha \end{array}}\right) \left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \sum _{\beta =\alpha }^{k}\left( {\begin{array}{c}\ell +\alpha -1\\ \beta \end{array}}\right) \left( {\begin{array}{c}m-\alpha \\ k-\beta \end{array}}\right) \\&=\sum _{\alpha =0}^{m}\left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \sum _{\beta =\alpha }^{\ell +\alpha -1}\left( {\begin{array}{c}\ell +\alpha -1\\ \beta \end{array}}\right) \sum _{k=\beta }^{\ell + \alpha -1}\left( {\begin{array}{c}\ell -1\\ k-\alpha \end{array}}\right) \left( {\begin{array}{c}m-\alpha \\ k-\beta \end{array}}\right) \\&=\sum _{\alpha =0}^{m}\left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \sum _{\beta =\alpha }^{\ell +\alpha -1}\left( {\begin{array}{c}\ell +\alpha -1\\ \beta \end{array}}\right) \sum _{k=\beta }^{\ell + \alpha -1}\left( {\begin{array}{c}\ell -1\\ \ell -1+\alpha -\beta -(k-\beta )\end{array}}\right) \left( {\begin{array}{c}m-\alpha \\ k-\beta \end{array}}\right) \\&=\sum _{\alpha =0}^{m}\left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \sum _{\beta =\alpha }^{\ell +\alpha -1}\left( {\begin{array}{c}\ell +\alpha -1\\ \beta \end{array}}\right) \left( {\begin{array}{c}\ell +m-\alpha -1\\ \ell +\alpha -\beta -1\end{array}}\right) \\&=\sum _{\alpha =0}^{m}\left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \sum _{\beta '=0}^{\ell -1}\left( {\begin{array}{c}\ell +\alpha -1\\ \beta '\end{array}}\right) \left( {\begin{array}{c}\ell +m-\alpha -1\\ \beta '\end{array}}\right) . \end{aligned}

$$\square$$

### Remark 4.2

Let $$G= K_\ell +E_m$$. Since

\begin{aligned}\sum _{\alpha =0}^{m}\left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \sum _{\beta =0}^{\ell +\alpha -1}\left( {\begin{array}{c}\ell +\alpha -1\\ \beta \end{array}}\right) \left( {\begin{array}{c}\ell +m-\alpha -1\\ \beta \end{array}}\right) =\left( {\begin{array}{c}2(\ell +m-1)\\ \ell +m-1\end{array}}\right) ,\end{aligned}

we have

\begin{aligned}{{\,\mathrm{Vol}\,}}\bigl (\nabla _G^PQ \bigr ) =\left( {\begin{array}{c}2(\ell +m-1)\\ \ell +m-1\end{array}}\right) -\sum _{\alpha =1}^{m-1}\left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \sum _{\beta =\ell }^{\ell +\alpha -1}\left( {\begin{array}{c}\ell +\alpha -1\\ \beta \end{array}}\right) \left( {\begin{array}{c}\ell +m-\alpha -1\\ \beta \end{array}}\right) .\end{aligned}

### Example 4.3

For small $$\ell$$ and m, $$f_{\ell , m}(1)$$ in Theorem 4.1 is

\begin{aligned} f_{1,m}(1)&=2^m,\\ f_{2,m}(1)&= 2^{m-2}(m^2+3 m+8),\\ f_{3,m}(1)&=\frac{2^{m-4}}{(2!)^2}(m^4+10 m^3+59 m^2+186 m+384),\\ f_{4,m}(1)&=\frac{2^{m-6}}{(3!)^2}(m^6+21 m^5+229 m^4+1563 m^3+7762 m^2+24{,}984 m+46{,}080),\\ f_{\ell ,1}(1)&=\left( {\begin{array}{c}2\ell \\ \ell \end{array}}\right) ,\\ f_{\ell ,2}(1)&=\left( {\begin{array}{c}2(\ell +1)\\ \ell +1\end{array}}\right) -2,\\ f_{\ell ,3}(1)&=\left( {\begin{array}{c}2(\ell +2)\\ \ell +2\end{array}}\right) -(6\ell +6),\\ f_{\ell ,4}(1)&=\left( {\begin{array}{c}2(\ell +3)\\ \ell +3\end{array}}\right) -(10\ell ^2+24\ell +20). \end{aligned}

From Theorems 1.1 and 4.1 we have the following.

### Corollary 4.4

Let G be a complete multipartite graph $$K_{m_1,\ldots ,m_s}$$, and let $$m= \sum _{i=1}^s m_i$$. Then we have

\begin{aligned} h^*\bigl (\nabla _G^PQ ,x\bigr )&=\sum _{i=1}^s f_{m-m_i,m_i}(x)- (s-1)\sum _{k=0}^{m -1}\left( {\begin{array}{c}m-1\\ k\end{array}}\right) ^{\!2}x^k,\\ {{\,\mathrm{Vol}\,}}\bigl (\nabla _G^PQ \bigr )&=\sum _{i=1}^s f_{m-m_i,m_i}(1) - (s-1)\left( {\begin{array}{c}2(m-1)\\ m-1\end{array}}\right) . \end{aligned}

### Example 4.5

From Corollary 4.4, the normalized volume of $$\nabla _{K_{\ell ,m}}^PQ$$ is

\begin{aligned}f_{\ell ,m}(1)+f_{m,\ell }(1) -\left( {\begin{array}{c}2(\ell +m-1)\\ \ell + m-1\end{array}}\right) .\end{aligned}

From Example 4.3 we have

\begin{aligned} {{\,\mathrm{Vol}\,}}\bigl (\nabla _{K_{1,m}}^PQ \bigr )&=2^m,\\ {{\,\mathrm{Vol}\,}}\bigl (\nabla _{K_{2,m}}^PQ \bigr )&=2^{m-2}(m^2+3 m+8) -2,\\ {{\,\mathrm{Vol}\,}}\bigl (\nabla _{K_{3,m}}^PQ \bigr )&=\frac{2^{m-4}}{(2!)^2}(m^4+10 m^3+59 m^2+186 m+384)-(6m+6),\\ {{\,\mathrm{Vol}\,}}\bigl (\nabla _{K_{4,m}}^PQ \bigr )&=\frac{2^{m-6}}{(3!)^2}(m^6+21 m^5+229 m^4+1563 m^3+7762 m^2+24984 m+46080)\\&\qquad \qquad \qquad -(10 m^2+24m +20). \end{aligned}

The formula for $${{\,\mathrm{Vol}\,}}\bigl (\nabla _{K_{2,m}}^PQ \bigr )$$ coincides with that in [8, Prop. 4.2].

### Example 4.6

Let G be the complete bipartite graph $$K_{2,n-2}$$. Since

\begin{aligned} f_{2,m}(x)&=\sum _{k=0}^{m+1}\sum _{\alpha =0}^{k}\left( {\begin{array}{c}1\\ k-\alpha \end{array}}\right) \left( {\begin{array}{c}m\\ \alpha \end{array}}\right) \sum _{\beta =\alpha }^{k}\left( {\begin{array}{c}\alpha +1\\ \beta \end{array}}\right) \left( {\begin{array}{c}m-\alpha \\ k-\beta \end{array}}\right) x^k\\&=\sum _{k=0}^{m+1}\left( {\begin{array}{c}m\\ k-1\end{array}}\right) k(m-k+1) x^k+\sum _{k=0}^{m+1}\left( {\begin{array}{c}m\\ k-1\end{array}}\right) x^k+\sum _{k=0}^{m+1}\left( {\begin{array}{c}m\\ k\end{array}}\right) (k+1) x^k\\&=\sum _{k=0}^{m}\left( {\begin{array}{c}m\\ k\end{array}}\right) k^2x^k+\sum _{k=0}^{m}\left( {\begin{array}{c}m\\ k\end{array}}\right) x^{k+1}+\sum _{k=0}^{m}\left( {\begin{array}{c}m\\ k\end{array}}\right) (k+1) x^k\\&=\sum _{k=0}^{m}\left( {\begin{array}{c}m\\ k\end{array}}\right) (x+1+k^2+k) x^k\\&=(x+1)^{m+1} + m( (m + 1) x +2) x(x+1)^{m-2} \end{aligned}

and

\begin{aligned}f_{\ell ,2}(x)=\sum _{k=0}^{\ell +1}\left( {\begin{array}{c}\ell +1\\ k\end{array}}\right) ^{\!2}x^k-2x,\end{aligned}

we have

\begin{aligned} h^*\bigl (\nabla _G^PQ ,x\bigr )&=f_{2,n-2}(x) + f_{n-2,2}(x) - \sum _{k=0}^{n -1}\left( {\begin{array}{c}n-1\\ k\end{array}}\right) ^{\!2}x^k\\&=(x+1)^{n-1} + (n-2)( (n- 1) x +2) x (x+1)^{n-4} -2x. \end{aligned}

## 5 Wheel Graphs

For $$n\ge 3$$, the wheel graph $$W_n$$ with $$n+1$$ vertices is the join graph $$W_n = C_n + K_1$$. Unfortunately, Theorem 1.1 is not useful for computing the $$h^*$$-polynomial of $$\nabla _{W_n}^PQ$$. We will give an explicit formula for the $$h^*$$-polynomial of $$\nabla _{W_n}^PQ$$ and prove the conjecture [8, Conj. 4.4] on the normalized volume of $$\nabla _{W_n}^PQ$$ by using Proposition 2.6 on $$\nabla _{G+K_1}^PQ$$. Let

\begin{aligned}\gamma (n,x)={\left\{ \begin{array}{ll} 1 &{} \quad \text {if}\;n=0,\\ \displaystyle \frac{(1+\sqrt{1+8x})^n + (1-\sqrt{1+8x})^n}{2^n} &{}\quad \text {if}\;n\text { is odd},\\ \displaystyle \frac{(1+\sqrt{1+8x})^n + (1-\sqrt{1+8x})^n}{2^n} -2 x^{n/2}&{}\quad \text {otherwise}. \end{array}\right. }\end{aligned}

For $$n\ge 3$$,

\begin{aligned}\frac{(1+\sqrt{1+8x})^n + (1-\sqrt{1+8x})^n}{2^n} = g(C_n, 2x),\end{aligned}

where $$g(C_n,x)$$ is the matching generating polynomial of $$C_n$$ ($$=$$ the “independence polynomial” of $$C_n$$). See, e.g., [18, p. 27]. Moreover, for $$n\ge 3$$, it is known [16, Exam. 4.5] that, $$\gamma (n,x)$$ is the $$\gamma$$-polynomial of the PV-type adjacency polytope $$\nabla ^PV _{W_n}$$ of $$W_n$$. (Note that $$\nabla ^\mathrm{PV}_{W_n}$$ is called the symmetric edge polytope of type A of $$W_n$$ in [16].) The $$h^*$$-polynomial of $$\nabla _{W_n}^PQ$$ is described by this function as follows.

### Theorem 5.1

The $$h^*$$-polynomial of $$\nabla _{W_n}^PQ$$ is

\begin{aligned} \sum _{k=0}^n \left( {\begin{array}{c}n\\ k\end{array}}\right) \gamma (n-k,x) x^k&=\biggl (\frac{1+2x+\sqrt{1+8x}}{2}\biggr )^{\!n}+\biggl (\frac{1+2x-\sqrt{1+8x}}{2}\biggr )^{\!n} \\&\quad +x^n-(x+\sqrt{x})^n-(x-\sqrt{x})^n. \end{aligned}

Moreover, the normalized volume of $$\nabla _{W_n}^PQ$$ is $$3^n-2^n+1$$.

### Proof

Since $$W_n = C_n + K_1$$, the $$h^*$$-polynomial of $$\nabla _{W_n}^PQ$$ is

\begin{aligned}p(D(C_n),x) =\sum _{\ell = 0}^n|PM (D(C_n),\ell )| x^\ell \end{aligned}

by Proposition 2.6. From Proposition 2.8, $$|PM (D(C_n),\ell )|$$ is equal to the number of (0, 1)-vectors $$(x_1,\ldots ,x_n, y_1,\ldots ,y_n) \in {\mathbb R}^{2n}$$ such that

\begin{aligned} \sum _{i =1}^{n} x_i&=\sum _{j=1}^{n} y_j = \ell , \end{aligned}
(14)
\begin{aligned} \sum _{i \in S} x_i&\,\le \!\sum _{\overline{j} \in \Gamma _{D(C_n)}(S)} \!\!y_j \quad \text{ for } \text{ all }\quad S \subset [n], \end{aligned}
(15)

where $$V= [n] \cup [\overline{n}]$$ is the set of vertices of $$D(C_n)$$. Let $$C_n=(1,2,\ldots ,n,1)$$. Given a subset $$T \subset [n]$$ and an integer $$\ell \in [n]$$, let $$PM _{T,\ell }$$ denote the set of all (0, 1) vectors $$(x_1,\ldots ,x_n,y_1,\ldots ,y_n) \in {\mathbb R}^{2n}$$ satisfying (14), (15), and $$T = \{ i \in [n] : x_i = y_i =1 \}$$. Note that $$PM _{T,\ell } = \emptyset$$ if $$\ell < |T|$$.

Let $$T \subset [n]$$ with $$|T| =k$$. We will show that $$\sum _{\ell =k}^n |PM _{T,\ell }| x^\ell = \gamma (n-k,x) x^k$$.

Case 1  ($$k=n$$). It is easy to see that $$PM _{T,n}= \{(1,\ldots ,1)\}$$ and $$PM _{T,\ell }=\emptyset$$ if $$\ell \ne n$$. Note that $$\gamma (n-k,x) x^k =x^n$$ if $$k=n$$. Thus $$\sum _{\ell =k}^n |PM _{T,\ell }| x^\ell = \gamma (n-k,x) x^k$$.

Case 2  ($$k=n-1$$). Let $$T=[n] \setminus \{i\}$$ where $$i \in [n]$$. It then follows that $$PM _{T,n-1}=\{ (1,\ldots ,1) - \mathbf{e}_i - \mathbf{e}_{n+i}\}$$ and $$PM _{T,\ell }=\emptyset$$ if $$\ell \ne n-1$$. Note that $$\gamma (n-k,x) x^k =x^{n-1}$$ if $$k=n-1$$. Thus $$\sum _{\ell =k}^n |PM _{T,\ell }| x^\ell = \gamma (n-k,x) x^k$$.

Case 3  ($$k=n-2$$). Let $$T=[n] \setminus \{i,j\}$$ where $$1 \le i<j\le n$$. Since (14) holds, each element of $$PM _{T,\ell }$$ is

\begin{aligned}\alpha _1 = (1,\ldots ,1) - \mathbf{e}_i - \mathbf{e}_j - \mathbf{e}_{n+i}- \mathbf{e}_{n+j}\end{aligned}

if $$\ell =n-2$$, and is one of

\begin{aligned}\alpha _2= (1,\ldots ,1) - \mathbf{e}_i - \mathbf{e}_{n+j},\qquad \alpha _3=(1,\ldots ,1) - \mathbf{e}_j - \mathbf{e}_{n+i},\end{aligned}

if $$\ell = n-1$$. Then each $$\alpha _i$$ corresponds to a perfectly matchable set. In fact, a matching of $$D(C_n)$$ which corresponds to $$\alpha _1,\alpha _2,\alpha _3$$ is

\begin{aligned}&\{ \{s, \overline{s}\} : s \in [n] \setminus \{i,j\} \},\\&\{ \{s, \overline{s}\} : 1 \le s \le i-1\; \text {or}\; j+1 \le s \le n \}\cup \{ \{ s+1 , \overline{s}\} : i \le s \le j-1 \},\\&\{ \{s, \overline{s}\} : 1 \le s \le i-1 \;\text {or}\; j+1 \le s \le n \}\cup \{ \{ s , \overline{s+1}\} : i \le s \le j-1 \}, \end{aligned}

respectively. Thus

\begin{aligned}PM _{T,\ell }={\left\{ \begin{array}{ll}\{\alpha _1\} &{}\quad \text {if}\;\ell =n-2,\\ \{\alpha _2, \alpha _3\} &{} \quad \text {if}\;\ell =n-1,\\ \emptyset &{} \quad \text {otherwise}.\end{array}\right. }\end{aligned}

Note that $$\gamma (n-k,x) x^k = (2x+1) x^{n-2} =x^{n-2} + 2 x^{n-1}$$ if $$k=n-2$$. Thus

\begin{aligned}\sum _{\ell =k}^n |PM _{T,\ell }| x^\ell = \gamma (n-k,x) x^k.\end{aligned}

Case 4  ($$k\le n-3$$). Suppose that $$T'=\{p,p+1,\ldots ,q\} \subset T$$ and $$p-1,q+1\notin T$$. Since $$|T|\le n-3$$, we have $$n-q+p-1\ge 3$$. Let $$U=\{ p,p+1,\ldots ,q, \overline{p}, \overline{p+1},\ldots ,\overline{q}\}$$. We will show that there exists a perfectly matchable set S of $$D(C_{n-q+p-1})$$ such that

\begin{aligned}\rho (S) =(x_1, \ldots , x_{p-1},x_{q+1},\ldots ,x_n, y_1, \ldots , y_{p-1},y_{q+1},\ldots ,y_n)\end{aligned}

for any $$(x_1,\ldots ,x_n,y_1,\ldots ,y_n) \in PM _{T,\ell }$$. Here $$C_{n-q+p-1}=(1,\ldots ,p-1,q+1,\ldots ,n,1)$$ and the vertex set of $$D(C_{n-q+p-1})$$ is $$\{1,\ldots ,p-1,q+1,\ldots ,n\} \cup \{\overline{1},\ldots ,\overline{p-1},\overline{q+1},\ldots ,\overline{n}\}$$. Let M be a matching of $$D(C_n)$$ which corresponds to $$(x_1,\ldots ,x_n,y_1,\ldots ,y_n)$$.

Case 4.1 (either $$x_{p-1}=x_{q+1}=0$$ or $$y_{p-1}=y_{q+1}=0$$). Exchanging [n] and $$[\overline{n}]$$ if needed, we may assume that $$y_{p-1}=y_{q+1}=0$$. Then, for the subset $$T' \subset [n]$$,

\begin{aligned}\sum _{i \in T'} x_i \,=\! \sum _{\overline{j} \in \Gamma _{G'}(T')}\! y_j = |T'|.\end{aligned}

Hence the matching M is the union of a perfect matching of the induced subgraph $$D(C_n)_U$$ of $$D(C_n)$$ and a matching $$M'$$ of the induced subgraph $$D(C_n)_{V\setminus U}$$ of $$D(C_n)$$. Then one can regard $$M'$$ as a matching of $$D(C_{n-q+p-1})$$ since $$D(C_n)_{V\setminus U}$$ is a subgraph of $$D(C_{n-q+p-1})$$.

Case 4.2  ($$x_{p-1}=y_{q+1} \ne y_{p-1}=x_{q+1}$$). Exchanging [n] and $$[\overline{n}]$$ if needed, we may assume that $$x_{p-1}=y_{q+1} =1$$ and $$y_{p-1}=x_{q+1}=0$$. If $$e = \{q+2, \overline{q+1} \}$$ belongs to the matching M, then $$M \setminus \{e\}$$ is the union of a perfect matching of $$D(C_n)_U$$ and a matching $$M'$$ of $$D(C_n)_{V\setminus U}$$ by the same argument in Case 4.1. Suppose that $$e = \{q, \overline{q+1} \}$$ belongs to the matching M. It then follows that M is the union of $$\{ \{p-1, \overline{p} \} , \ldots , \{q-1, \overline{q} \} , \{q, \overline{q+1}\}\}$$ and a matching $$M'$$ of $$D(C_n)_{V\setminus U}$$. Thus one can regard $$M' \cup \{\{p-1, \overline{q+1} \} \}$$ as a matching of $$D(C_{n-q+p-1})$$.

Thus there exists a perfectly matchable set $$S_1$$ of $$D(C_{n-q+p-1})$$ such that

\begin{aligned}\rho (S_1) =(x_1, \ldots , x_{p-1},x_{q+1},\ldots ,x_n, y_1, \ldots , y_{p-1},y_{q+1},\ldots ,y_n)\end{aligned}

for $$(x_1,\ldots ,x_n,y_1,\ldots ,y_n)$$ if $$T'=\{p,p+1,\ldots ,q\} \subset T$$ and $$p-1 , q+1 \notin T$$. If, in addition, $$T''=\{p',p'+1,\ldots ,q'\} \subset T$$ and $$p'-1 , q'+1 \notin T$$ for some $$p' > q+1$$, then there exists a perfectly matchable set $$S_2$$ of $$D(C_{n-(q-p+1) -(q'-p'+1)})$$ such that

\begin{aligned}\rho (S_2)= & {} (x_1, \ldots , x_{p-1},x_{q+1},\ldots , x_{p'-1},x_{q'+1},\ldots , x_n, \\&y_1, \ldots , y_{p-1},y_{q+1},\ldots , y_{p'-1},y_{q'+1},\ldots ,y_n) \end{aligned}

by the same argument as above. Repeating the above argument, it follows that there exists a perfectly matchable set S of $$D(C_{n-k})$$ such that $$\rho (S) =(x_{i_1}, \ldots , x_{i_{n-k}}, y_{i_1}, \ldots , y_{i_{n-k}})$$ where $$[n] \setminus T = \{i_1,\ldots ,i_{n-k}\}$$ and $$x_{i_r} + y_{i_r} \le 1$$ for all r. Then there exists a perfectly matchable set $$S'$$ of $$C_{n-k}$$ such that $$|S'| = 2(\ell -k)$$ and $$\rho (S') = (x_{i_1} +y_{i_1}, \ldots , x_{i_{n-k}} + y_{i_{n-k}})$$. The matching corresponding to $$S'$$ is not unique if and only if $$n-k$$ is even and $$\rho (S')=(1,\ldots ,1)$$. There exist two matchings for the perfectly matchable set S of $$D(C_{n-k})$$ exactly when

\begin{aligned}(x_{i_1}, \ldots , x_{i_{n-k}}, y_{i_1}, \ldots , y_{i_{n-k}})= & {} (1,0,\ldots ,1,0, 0,1,\ldots , 0,1), \\&(0,1,\ldots ,0,1,1,0, \ldots , 1,0).\end{aligned}

Conversely, for each matching M of $$C_{n-k}$$, there exist $$2^{|M|}$$ vectors $$(x_{i_1}, \ldots , x_{i_{n-k}}, y_{i_1}, \ldots , y_{i_{n-k}})$$ where $$x_{i_r} + y_{i_r} \le 1$$ for all r and associated with a perfectly matchable set of $$D(C_{n-k})$$ since there are two possibilities $$(x_{i_p}, x_{i_{p+1}},y_{i_p}, y_{i_{p+1}}) =(1,0,0,1), (0,1,1,0)$$ for each $$\{i_p, i_{p+1}\} \in M$$. Hence we have

\begin{aligned}| \mathrm{PM}_{T,\ell } | ={\left\{ \begin{array}{ll} 2^{\ell -k} m_{C_{n-k}} (\ell -k) -2 &{}\quad \text {if}\;\ell -k = (n-k)/2,\\ 2^{\ell -k} m_{C_{n-k}} (\ell -k)&{} \quad \text {otherwise}. \end{array}\right. }\end{aligned}

Recall that $$m_{G} (k)$$ is the number of k-matchings of a graph G. Thus

\begin{aligned} \sum _{\ell =k}^n |PM _{T,\ell }| x^\ell&=\sum _{\ell =k}^n2^{\ell -k} m_{C_{n-k}} (\ell -k) x^\ell -2 x^{(n+k)/2} \\&=\left( \sum _{\ell =k}^n m_{C_{n-k}} (\ell -k)(2 x)^{\ell -k} -2 x^{(n-k)/2} \right) x^k=\gamma (n-k,x) x^k \end{aligned}

if $$n-k$$ is even, and

\begin{aligned} \sum _{\ell =k}^n |PM _{T,\ell }| x^\ell&=\sum _{\ell =k}^n2^{\ell -k} m_{C_{n-k}} (\ell -k) x^\ell \\&=\left( \sum _{\ell =k}^nm_{C_{n-k}} (\ell -k)(2 x)^{\ell -k}\right) x^k=\gamma (n-k,x) x^k \end{aligned}

if $$n-k$$ is odd. Therefore the $$h^*$$-polynomial of $$\nabla _{W_n}^PQ$$ is

\begin{aligned}\sum _{T \subset [n]}\sum _{\ell =|T|}^n |PM _{T,\ell }| x^\ell =\sum _{T \subset [n]}\gamma (n-|T|) x^{|T|}=\sum _{k=0}^n \left( {\begin{array}{c}n\\ k\end{array}}\right) \gamma (n-k,x) x^k.\end{aligned}

Moreover,

\begin{aligned}&\sum _{k=0}^n \left( {\begin{array}{c}n\\ k\end{array}}\right) \gamma (n-k,x) x^k\\&\quad =x^n+\sum _{k=0}^n \left( {\begin{array}{c}n\\ k\end{array}}\right) \left( \frac{1+\sqrt{1+8x}}{2}\right) ^{n-k} x^k+\sum _{k=0}^n \left( {\begin{array}{c}n\\ k\end{array}}\right) \biggl (\frac{1-\sqrt{1+8x}}{2}\biggr )^{\!n-k}x^k\\&\qquad - \sum _{\begin{array}{c} k\ge 0\\ k \equiv n \; (\mathrm{mod}\;2) \end{array}}\!\!\! 2 \left( {\begin{array}{c}n\\ k\end{array}}\right) \sqrt{x}^{n-k}\,x^k\\&\quad =\biggl (\frac{1+2x+\sqrt{1+8x}}{2}\biggr )^{\!n} \!+\!\biggl (\frac{1+2x-\sqrt{1+8x}}{2}\biggr )^{\!n}\!+\!x^n-(x\!+\!\sqrt{x})^n\!-\! (x-\sqrt{x})^n. \end{aligned}

In particular, substituting $$x=1$$, we have

\begin{aligned}\sum _{k=0}^n \left( {\begin{array}{c}n\\ k\end{array}}\right) \gamma (n-k,1)=3^n-2^n+1. \end{aligned}

$$\square$$