PQ-Type Adjacency Polytopes of Join Graphs

PQ-type adjacency polytopes ∇PQG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla ^PQ _G$$\end{document} are lattice polytopes arising from finite graphs G. There is a connection between ∇PQG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla ^PQ _G$$\end{document} 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 ∇PQG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla ^PQ _G$$\end{document} 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∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^*$$\end{document}-polynomial and the normalized volume of ∇PQG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla ^PQ _G$$\end{document} of a join graph G are presented. Moreover, we give explicit formulas of the h∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$h^*$$\end{document}-polynomial and the normalized volume of ∇PQG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla ^PQ _G$$\end{document} when G is a complete multipartite graph or a wheel graph.


INTRODUCTION
A lattice polytope P ⊂ R n is a convex polytope all of whose vertices have integer coordinates.Its normalized volume, Vol(P) = dim(P)!vol(P)where vol(P) is the relative volume of P, is always a positive integer.To compute Vol(P) is a fundamental but hard problem in polyhedral geometry.
Let G be a simple graph on [n] := {1, . . ., n} with edge set E(G).The PV-type adjacency polytope ∇ PV G of G is the lattice polytope which is the convex hull of {±(e i − e j ) ∈ R n : {i, j} ∈ E(G)}, where e i is the i-th unit coordinate vector in 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 ∇ PQ G of G is the lattice polytope which is the convex hull of {(e i , e j ) ∈ R 2n : {i, j} ∈ E(G) or i = j}.
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.Davis and Chen [8] showed that the normalized volume of ∇ PQ G can be computed by using sequences of nonnegative integers related to the Dragon Marriage Problem [17].
In the present paper, we focus on the h * -polynomial of a PQ-type adjacency polytope.Here, the h * -polynomial h * (P, x) of a lattice polytope P is a discrete tool to compute the normalized volume Vol(P) (see Section 2).We recall a relation between ∇ PQ G and a root polytope.For a bipartite graph H on [n] with edge set E(H), the root polytope Q H of H is the lattice polytope which is the convex hull of {e i + e j ∈ R n : {i, j} ∈ E(H)}.
For a positive integer n, set [n] := {1, . . ., n}.Define D(G) to be the bipartite graph on [n] ∪ [n] with edges {i, i} for each i ∈ [n] and {i, j} and {i, j} for each edge {i, j} in G.It then follows that Lemma 2.4]).On the other hand, it is known [12] that the h * -polynomial of the root polytope 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 Q H is equal to |HT(H)|, where HT(H) denotes the set of hypertrees of the associated hypergraph of H. Therefore, we can compute the h * -polynomial and the normalized volume of ∇ PQ G of a connected graph G by using an interior polynomial and counting hypertrees.
The main result of the present paper is formulas of the h * -polynomial and the normalized volume of ∇ PQ G of a join graph G. Let G 1 , . . ., G s be graphs with m 1 , . . ., m s vertices.Suppose that G i and G j have no common vertices for each i = j.Then the join G For example, the complete bipartite graph K ,m is equal to the join E + E m where E k is the empty graph with k vertices.For complete graphs K and K m , one has K + K m = K +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 Section 2 for the definition of perfectly matchable set polynomials).
Theorem 1.1.Let G 1 , . . ., G s be graphs with m 1 , . . ., m s vertices.Suppose that G i and G j have no common vertices for each i = j.Then for the join G 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 ∇ 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 ∇ 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 ∇ PQ W n and prove the conjecture [8,Conjecture 4.4] on the normalized volume of ∇ PQ W n (Theorem 5.1) by computing the perfectly matchable set polynomial of 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 Section 2, we give a proof of Theorem 1.1 in Section 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 Section 4. Finally, we compute the h * -polynomial and the normalized volume of the PQ-type adjacency polytope of a wheel graph in Section 5.

PRELIMINARIES
As explained in the previous section, the h * -polynomial of ∇ PQ G is equal to the interior polynomial of D(G).First, we recall what h * -polynomials are.Let P ⊂ R n be a lattice polytope of dimension d.Given a positive integer t, we define where tP := {tx ∈ R n : x ∈ P}.The study on L P (t) originated in Ehrhart [9] who proved that L P (t) is a polynomial in t of degree d with the constant term 1.We call L P (t) the Ehrhart polynomial of P. The generating function of the lattice point enumerator, i.e., the formal power series is called the Ehrhart series of P. It is known that it can be expressed as a rational function of the form , where h * (P, x) is a polynomial in x of degree at most d with nonnegative integer coefficients and it is called the h * -polynomial (or the δ -polynomial) of P.Moreover, , where int(P) is the relative interior of P. Furthermore, h * (P, 1) = ∑ d i=0 h * i is equal to the normalized volume Vol(P) of P. We refer the reader to [4] for the detailed information about Ehrhart polynomials and h *polynomials.
Next, we recall the definition of interior polynomials and their properties.A hypergraph is a pair H = (V, E), where E = {e 1 , . . ., e n } is a finite multiset of non-empty subsets of V = {v 1 , . . ., v m }.Elements of V are called vertices and the elements of E are the hyperedges.Then we can associate H to a bipartite graph BipH on the vertex set V ∪ E with the edge set {{v i , e j } : v i ∈ e j }.Assume that BipH is connected.A hypertree in H is a function f : E → Z ≥0 such that there exists a spanning tree Γ of BipH whose vertices have degree f (e) + 1 at each e ∈ E. Then we say that Γ induces f .Let HT(H ) denote the set of all hypertrees in H .A hyperedge e j ∈ E is said to be internally inactive with respect to the hypertree f if there exists j < j such that g : E → Z ≥0 defined by (otherwise) is a hypertree.Let ι( f ) be the number of internally inactive hyperedges of f .Then the interior polynomial of H is the generating function The coefficients of I G (x) are described as follows.
the following condition: there exists j < j such that the function g satisfies condition (ii) above.
From [8, Lemma 2.4] and [12, Theorems 1.1 and 3.10], we have the following.[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 / 0 as a perfectly matchable set and we set PM(G, 0 the perfectly matchable set polynomial (PMS polynomial) of G.

Assume that G is a bipartite graph with a bipartition
Proposition 2.4 ([14, Proposition 3.4]).Let G be a bipartite graph.Then we have The following Lemma is easy to see.However it is useful when we apply Proposition 2.4 to such graphs.Lemma 2.5.Let G be a graph.Then we have Moreover, we have Proposition 2.6.Let G be a graph with n vertices.Then the h * -polynomial of ∇ PQ G+K 1 is p(D(G), x).In particular, the normalized volume of Proof.The assertion follows from Propositions 2.2, 2.4 and Lemma 2.5.
Remark 2.7.Given a graph G, the number of matchings of G is called Hosoya index of G and denoted by Z(G).From Proposition 2.6, the normalized volume of ∇ PQ G+K 1 is at most Z(D(G)).
Let G be a graph on the vertex set [n].Given a subset S ⊂ [n], we associate the (0, 1)-vector ρ(S) = ∑ i∈S e i ∈ R n .For example, ρ( / 0) = 0 ∈ R n .The convex hull of {ρ(S) : S ∈ PM(G, k) for some k} 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, Theorem 1]).Let G be a bipartite graph on the vertex set where Γ G (S) ⊂ V 2 is the set of vertices adjacent to some vertex in S.

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 . By the similar argument as in Case 2, if f satisfies condition (i), then condition (ii) in Proposition 2.1 is equivalent to the condition Hence, if f satisfies condition (i), then condition (ii) in Proposition 2.1 holds if and only if both ( 6) and ( 7) hold.
Since any element in Suppose that f satisfies none of ( 6) and (7).Then It then follows that Hence, if f satisfies condition (i), then at least one of ( 6) or ( 7) holds.Thus (5) holds.
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 as in Fig. 1.
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 where η is defined in Proposition 2.1).In order to prove k = , we will show that η (6) 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 η D(G 1 +K m 2 ) ( f ).We now show that any v j ( j ≥ m 1 + 2) belongs to η D(G) ( f ), i.e., there exists j < j such that g : V 1 → Z ≥0 defined by (9) g is a hypertree in D(G).Let Γ be a spanning tree of D(G) that induces f .Suppose that Γ does not contain an edge e = {v k , v 1 } for some m 1 < k ≤ m.Then Γ ∪ {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 (Γ ∪ {e }) \ {e} induces f .Thus we may assume that {{v k , v 1 } : m 1 < k ≤ m} ⊂ Γ.By the same argument, we may assume that Since Γ has 2m − 1 edges, we have Γ = S T {e} for some edge e of D(G).In particular, the degree of each vertex v i (i ∈ [m] \ {1, m 1 + 1}) in Γ is at most 2. Since f (v j ) > 0, the degree of v j in Γ is at least 2, and hence {v j , v k } is an edge of Γ for some k ∈ [m] \ {1}.Let e = {v j , v k }.We now construct a spanning tree of D(G) that induces a hypertree of the form (9). Note that, for an edge e of D(G), Γ = (Γ ∪ {e }) \ {e } is a spanning tree of D(G) if e / ∈ Γ and the (unique) cycle of Γ ∪ {e } contains e .
that induces a hypertree of the form (9) where that induces a hypertree of the form (9) where j = 1.
Thus we have k = , and hence (10) In particular, and hence From ( 8) and (11), Finally, we show that this is a decomposition.Suppose that . From (4), f ∈ F G ( ) for some .Moreover, from (10), we have = k.Thus as in Fig. 1.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., Thus we have Let s > 2 and assume that the assertion holds for the join of at most s − 1 graphs.Since From Proposition 2.4 and Lemma 2.5, this is equal to equation (3).

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 and m, let holds, we have Thus (12) and ( 13) hold if and only if We count such vectors with |{i ∈ V 1,2 : x i = 1}| = α for each α = 0, 1, . . ., k.There are m α possibilities for the choice of the subset S = {i ∈ V 1,2 : For each S , there are −1 k−α possibilities for the choice of the subset {i ∈ V 1,1 : Then equations ( 14) and ( 15 Moreover, the normalized volume of ∇ PQ G is equal to Example 4.3.For small and m, f ,m (1) in Theorem 4.1 is From Theorems 1.1 and 4.1, we have the following.

WHEEL GRAPHS
For n ≥ 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 ∇ PQ W n .We will give an explicit formula for the h * -polynomial of ∇ PQ W n and prove the conjecture [8,Conjecture 4.4] on the normalized volume of ∇ PQ W n by using Proposition 2.6 on ∇ PQ G+K 1 .Let For n ≥ 3, where g(C n , x) is the matching generating polynomial of C n .Moreover, for n ≥ 3, it is known [16, Example 4.5] that, γ(n, x) is the γ-polynomial of the PV-type adjacency polytope ∇ PV W n of W n .(Note that ∇ PV W n is called the symmetric edge polytope of type A of W n in [16].)The h * -polynomial of ∇ PQ W n is described by this function as follows.
Conversely, for each matching M of C n−k , there exist 2 |M| vectors (x i 1 , . . ., x i n−k , y i 1 , . . ., y i n−k ) where x i r + y i r ≤ 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 } ∈ M.

Corollary 4 . 4 .
Let G be a complete multipartite graph K m 1 ,...,m s , and let m = ∑ s i=1 m i .Then we have h