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.
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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
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
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
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
where \(p(H,x)\) denotes the perfectly matchable set polynomial of a graph H. In particular, one has
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
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
is called the Ehrhart series of \({\mathscr {P}}\). It is known that it can be expressed as a rational function of the form
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,
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
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
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
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
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
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
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
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
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
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
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
Suppose that f satisfies none of (4) and (5). Then
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
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
Therefore
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
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
is a subset of \(\Gamma \). Since \(\Gamma \) is spanning,
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
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
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
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
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
satisfies
for all \(\emptyset \ne V'' \subset V_{1,1}\). Hence g satisfies (4) in the proof of Lemma 3.1. Therefore we have
and hence
Claim 4
\(F_{K_m}(k)=F_{K_{m_1}+G_2}(k) \cup (F_{G_1+K_{m_2}}(k) \setminus F_G(k))\).
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
Then \(\ell =k\). It follows that
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
Therefore \(F_{K_m}(k)\) is decomposed into the disjoint sets
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
From Lemma 3.2,
Thus we have
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
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
Since
holds, we have
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
Proof
Let \(G'=D(K_{\ell -1}+E_m)\). Since \(G =(K_{\ell -1}+E_m) +K_1\), from Proposition 2.6, we have
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
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
Thus (10) and (11) hold if and only if
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
\(\square \)
Remark 4.2
Let \(G= K_\ell +E_m \). Since
we have
Example 4.3
For small \(\ell \) and m, \(f_{\ell , m}(1)\) in Theorem 4.1 is
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
Example 4.5
From Corollary 4.4, the normalized volume of \(\nabla _{K_{\ell ,m}}^PQ \) is
From Example 4.3 we have
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
and
we have
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
For \(n\ge 3\),
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
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
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
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
if \(\ell =n-2\), and is one of
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
respectively. Thus
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
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
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 \(T' \subset [n]\),
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
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
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
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
Recall that \(m_{G} (k)\) is the number of k-matchings of a graph G. Thus
if \(n-k\) is even, and
if \(n-k\) is odd. Therefore the \(h^*\)-polynomial of \(\nabla _{W_n}^PQ \) is
Moreover,
In particular, substituting \(x=1\), we have
\(\square \)
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Acknowledgements
The authors thank the anonymous referees for their careful reading and helpful suggestions. The authors were partially supported by JSPS KAKENHI 18H01134, 19K14505, and 19J00312.
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Ohsugi, H., Tsuchiya, A. PQ-Type Adjacency Polytopes of Join Graphs. Discrete Comput Geom 70, 214–235 (2023). https://doi.org/10.1007/s00454-022-00447-z
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DOI: https://doi.org/10.1007/s00454-022-00447-z
Keywords
- Adjacency polytope
- \(h^*\)-polynomial
- Interior polynomial
- Perfectly matchable set
- Join graph
- Root polytope