PQ-Type Adjacency Polytopes of Join Graphs

Abstract

PQ-type adjacency polytopes $\nabla ^PQ _G$ are lattice polytopes arising from finite graphs G. There is a connection between $\nabla ^PQ _G$ and the engineering problem known as power-flow study, which models the balance of electric power on a network of power generation. In particular, the normalized volume of $\nabla ^PQ _G$ plays a central role. In the present paper, we focus on the case where G is a join graph. In particular, formulas of the $h^*$-polynomial and the normalized volume of $\nabla ^PQ _G$ of a join graph G are presented. Moreover, we give explicit formulas of the $h^*$-polynomial and the normalized volume of $\nabla ^PQ _G$ when G is a complete multipartite graph or a wheel graph.

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

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

• (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;

• (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.

Fig. 1

Decomposition of $F_{K_m}(k)$

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

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

$\square$