Symmetric edge polytopes and matching generating polynomials

Symmetric edge polytopes $\mathcal{A}_G$ of type A are lattice polytopes arising from the root system $A_n$ and finite simple graphs $G$. There is a connection between $\mathcal{A}_G$ and the Kuramoto synchronization model in physics. In particular, the normalized volume of $\mathcal {A}_G$ plays a central role. In the present paper, we focus on a particular class of graphs. In fact, for any cactus graph $G$, we give a formula for the $h^*$-polynomial of $\mathcal{A}_{\widehat{G}}$ by using matching generating polynomials, where $\widehat{G}$ is the suspension of $G$. This gives also a formula for the normalized volume of $\mathcal{A}_{\widehat{G}}$. Moreover, via the chemical graph theory, we show that for any cactus graph $G$, the $h^*$-polynomial of $\mathcal{A}_{\widehat{G}}$ is real-rooted. Finally, we extend the discussion to symmetric edge polytopes of type $B$, which are lattice polytopes arising from the root system $B_n$ and finite simple graphs.


INTRODUCTION
A lattice polytope P ⊂ R n is a convex polytope all of whose vertices have integer coordinates. Many lattice polytopes arising from graphs are constructed and have been studied from several viewpoints.
In [24], lattice polytopes arising from the root system of type A n and finite graphs were introduced. Let G be a finite simple undirected graph on the vertex set [n] := {1, . . ., n} with the edge set E(G). The symmetric edge polytope A G (of type A) of G is the lattice polytope which is the convex hull of where e i is the i-th unit coordinate vector in R n . Symmetric edge polytopes, in particular, their Ehrhart polynomials were studied from a viewpoint of algebraic number theory. In fact, the Ehrhart polynomials of special symmetric edge polytopes have properties similar to Riemann's ζ function [5,29]. Moreover, many results about zero loci of the Ehrhart polynomials of symmetric edge polytopes have been found from a viewpoint of algebraic combinatorics [19,22,25]. On the other hand, symmetric edge polytopes are always reflexive polytopes, i.e., their dual polytopes are also lattice polytopes. Reflexive polytopes correspond to Gorenstein toric Fano varieties and they give many explicit constructions of mirrors of Calabi-Yau hypersurfaces [1]. By a work of Hibi [17], it follows that their h * -polynomials of reflexive polytopes are always palindromic. One of current hot topics on the study of lattice polytopes is unimodality questions for h * -polynomials. Since any symmetric edge polytope has a regular unimodular triangulation, its h * -polynomial is unimodal due to a result of Bruns and Römer [4]. Moreover, the h * -polynomial of A G coincides with the h-polynomial of a unimodular triangulation of the boundary ∂ A G . From a viewpoint of algebraic and topological combinatorics, Gal [12] conjectured that the h-polynomial of a flag triangulation of a sphere is γ-positive, which directly implies the unimodality. More strongly, Nevo and Petersen conjectured that the γ-polynomial of the h-polynomial of a flag triangulation of a sphere coincides with the f -polynomial of a balanced simplicial complex [23]. In [18,27], it is shown that the h * -polynomials of the symmetric edge polytopes of certain classes of graphs are γ-positive. On the other hand, the h * -polynomial of A G is not always real-rooted (Example 5.2). Note that if a polynomial all of whose coefficients are positive is palindromic and real-rooted, then it is γ-positive and log-concave.
Recently, the normalized volumes of symmetric edge polytopes have attracted much attention. The author of [6] calls symmetric edge polytopes adjacency polytopes and refer to the normalized volumes as the adjacency polytope bounds. Adjacency polytopes appeared in the context of the Kuramoto model, describing the behavior of interacting oscillators [21]. There are many applications of the Kuramoto model in several fields of study in biology, physics, chemistry, engineering, and social science. In many cases, an adjacency polytope bound gives an upper bound of the number of possible solutions in the Kuramoto equations [6]. In [7,8,25], explicit formulas of the adjacency polytope bounds, i.e., the normalized volumes of the symmetric edge polytopes of certain classes of graphs are given. Moreover, in [9], another type of adjacency polytope bounds is discussed.
In the present paper, from the above background we investigate the normalized volume and the γ-polynomial of the h * -polynomial of a symmetric edge polytope. In particular, we focus on the suspension G of a graph G. Here G is the graph on the vertex set [n + 1] and the edge set E(G) ∪ {{i, n + 1} : i ∈ [n]}. In [27], a formula of the γ-polynomial of the h * -polynomial of A G by using interior polynomials, which are a version of the Tutte polynomials for hypergraphs introduced by Kálmán [20], is given. Moreover, the h * -polynomial of A G is always γ-positive. Furthermore, this formula also gives a formula of the normalized volume of A G .
Our first main theorem is another formula of the γ-polynomial of the h * -polynomial of A G of a certain class of graphs G by using matching generating polynomials. A cactus graph is a graph G such that each edge of G belongs to at most one cycle of G. Theorem 1.1. Let G be a cactus graph on [n] and let g(G, x) be the matching generating polynomial of G. Then the γ-polynomial of the h * -polynomial of A G is where R ′ 2 (G) is the set of all subgraphs of G consisting of vertex-disjoint even cycles, and c(R) is the number of the cycles of R. Moreover, the normalized volume of A G is In [8], the normalized volume of the symmetric edge polytope of a wheel graph is computed. Note that a wheel graph is the suspension of a cactus graph. By using Theorem 1.1, 2 we can give explicit formulas of the normalized volume and the γ-polynomial of the h *polynomial of the symmetric edge polytope of a wheel graph (Example 4.5). On the other hand, applying Theorem 1.1 to a graph G which has no even cycle, we can describe the γ-polynomial of the h * -polynomial of A G by a single matching generating polynomial and we know that it is real-rooted (Corollary 4.4). Our second main theorem extends the real-rootedness of this result to the suspension of any cactus graph via methods from chemical graph theory. In fact, Theorem 1.2. Let G be a cactus graph. Then the h * -polynomial of A G is real-rooted.
In the present paper, we also discuss Nevo-Petersen's conjecture for A G . More precisely, we show the following. Theorem 1.3. Let G be a graph which has no even cycles. Then the γ-polynomial of h * (A G , x) coincides with the f -polynomial of a flag simplicial complex.
Note that the h-polynomial of a flag simplicial complex coincides with that of a balanced simplicial complex [11]. Hence Theorem 1.3 proves that Nevo-Petersen's conjecture holds for any flag unimodular triangulation of the boundary ∂ A G in this case. We also show that for a forest G, ∂ A G has a flag unimodular triangulation by using the algebraic technique of Gröbner bases from Theorem 6.9 (since a forest is a bipartite graph with no cycles).
In [26], lattice polytopes arising from the root system of type B n and finite graphs were introduced. The symmetric edge polytope B G of type B of a graph G on the vertex set [n] is the lattice polytope which is the convex hull of Then it follows that B G is reflexive if and only if G is a bipartite graph. In the case, B G has a regular unimodular triangulation. Moreover, if G is bipartite, the γ-polynomial of the h * -polynomial of B G can be described by an interior polynomial. Similarly to the case of symmetric edge polytopes of type A, we give a formula of the γ-polynomial of the h * -polynomial of B G for a cactus bipartite graph G by using matching generating polynomials and prove that the h * -polynomial is real-rooted (Theorem 7.2). Moreover, we show that for a forest G, the γ-polynomial of the h * -polynomial of B G coincides with the f -polynomial of a flag simplicial complex (Theorem 7.5). Namely, Nevo-Petersen's conjecture holds for any flag unimodular triangulation of the boundary ∂ B G in this case. We remark that for any forest G, ∂ B G has a flag unimodular triangulation (Remark 7.6).
The paper is organized as follows: In Section 2, we recall the definition of the h *polynomials of lattice polytopes, some properties of polynomials and their related conjectures. In Section 3, we define the interior polynomials of connected bipartite graphs and recall a formula of the γ-polynomial of the h * -polynomial of A G for a graph G in terms of interior polynomials. We give the proofs of Theorems 1.1, 1.2 and 1.3 in Sections 4, 5 and 6 respectively. Finally, in Section 7, we extend the discussion to symmetric edge polytopes of type B.

EHRHART THEORY AND γ-POLYNOMIALS
In this section, we recall the definition of h * -polynomials, the notion of γ-positivity and its related properties. Let P ⊂ R n be a lattice polytope of dimension d. Given a positive integer t, we define L P (t) = |tP ∩ Z n |, where tP := {tx ∈ R n : x ∈ P}. The study on L P (t) originated in Ehrhart [10] 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 is a polynomial in x of degree at most d with nonnegative integer coefficients [30] and it is called the h * -polynomial (or the δ -polynomial) of P. Moreover, where relint(P) is the relative interior of P. Furthermore, h * (P, 1) = ∑ d i=0 h * i is equal to the normalized volume of P. We refer the reader to [2] for the detailed information about Ehrhart polynomials and h * -polynomials.
A full-dimensional lattice polytope P ⊂ R n is called reflexive if the origin of R n belongs to the interior of P and its dual polytope is also a lattice polytope, where x, y is the usual inner product of R n . Two lattice polytopes P ⊂ R n and Q ⊂ R m are said to be unimodularly equivalent if there exists an affine map from the affine span aff(P) of P to the affine span aff(Q) of Q that maps Z n ∩ aff(P) bijectively onto Z m ∩ aff(Q) and maps P to Q. Each lattice polytope is unimodularly equivalent to a full-dimensional lattice polytope. In general, we say that a lattice polytope is reflexive if it is unimodularly equivalent to a reflexive polytope. We can characterize when a lattice polytope is reflexive in terms of its h * -polynomial. Let f = ∑ d i=0 a i x i be a polynomial with real coefficients and a d = 0. We now focus on the following properties.
(RR) We say that f is real-rooted if all its roots are real.
We say that f is unimodal if a 0 ≤ a 1 ≤ · · · ≤ a k ≥ · · · ≥ a d for some k. If all its coefficients are positive, then these properties satisfy the implications Assume that f is palindromic. Then f has a unique expression We can see that a γ-positive polynomial is real-rooted if and only if its γ-polynomial has only real roots [28,Observation 4.2]. For a reflexive polytope P, denote γ(P, x) the γ-polynomial of h * (P, x).
For a given lattice polytope, a fundamental problem within the field of Ehrhart theory is to determine if its h * -polynomial is unimodal. One famous instance is given by reflexive polytopes that possess a regular unimodular triangulation.
It is known that if a reflexive polytope possesses a flag regular unimodular triangulation all of whose maximal simplices contain the origin, then the h * -polynomial coincides with the h-polynomial of a flag triangulation of a sphere [4]. For the h-polynomial of a flag triangulation of a sphere, Gal conjectured the following:

INTERIOR POLYNOMIALS
In [27], for any graph G a formula for γ(A G , x) in terms of interior polynomials was given. In this section, we recall the definition of interior polynomials and the formula.
A hypergraph is a pair H = (V, E), where E = {e 1 , . . . , e n } is a finite multiset of nonempty 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 active with respect to the hypertree f if it is not possible to decrease f(e j ) by 1 and increase f(e j ′ ) ( j ′ < j) by 1 so that another hypertree results. We call a hyperedge internally inactive with respect to a hypertree if it is not internally active and denote the number of such hyperedges of f by ι(f). Then the interior polynomial of H is the generating Let G be a finite graph on [n] with the edge set E(G). Given a subset S ⊂ [n], We identify E S with the subgraph of G on the vertex set [n] and the edge set E S . By definition, E S is a bipartite graph. Let Cut(G) be the set of all cuts of G. Note that |Cut(G)| = 2 n−1 .
Assume that G is a bipartite graph with a bipartition V 1 ∪ V 2 = [n]. Then let G be a connected bipartite graph on [n + 2] whose edge set is In particular, In this section, we prove Theorem 1.1. First we recall a relation between interior polynomials and k-matchings. Let G be a finite graph with n vertices. A k-matching of G is a set of k pairwise non-adjacent edges of G. Let there exists a k-matching of G whose vertex set is where m k (G) is the number of k-matchings in G. On the other hand, the matching gener- It is known [13, Theorem 5.5.1] that α(G, x) is real-rooted for any graph G. Since α(G, x) = x n g(G, −x −2 ), it follows that any root of g(G, x) is real and negative. In fact, if u is a root of g(G, x), then u is not zero and (−u) −1/2 is a root of α(G, x). Thus v = (−u) −1/2 is real and hence u = −v −2 is real and negative. 6 Lemma 4.2. Let G be a graph such that each edge of G belongs to at most one even cycle of G. Then Since C belongs to EC(V ), there exists M 1 , M 2 ∈ M(V ) such that C appears in the graph G M 1 ,M 2 . Suppose that there exists j such that neither {v j−1 , v j } nor {v j , v j+1 } belongs to M. Then M has an edge {v j , w} with w / ∈ {v j−1 , v j+1 }. We may assume that {v j−1 , v j } belongs to M 1 . Then G M,M 1 has edges {v j−1 , v j } and {v j , w}. Hence there exists an even cycle of G that contains {v j−1 , v j } and {v j , w}, a contradiction. Thus one of {v j−1 , v j } or {v j , v j+1 } belongs to M. Hence the intersection of C and M is either as desired. Now, we prove Theorem 1.1. In fact, Theorem 1.1 follows from the following more general result.
Moreover, the normalized volume of A G is Since each edge of G belongs to at most one even cycle of G, each H ∈ Cut(G) satisfies the same condition. From Lemma 4.2, for each H ∈ Cut(G). Thus the γ-polynomial of A G is Note that every k-matching of H ∈ Cut(G) is a k-matching of G.
• Let M be a k-matching of G. Then M is a k-matching of H ∈ Cut(G) if and only if M is a subgraph of H. There are 2 n−k−1 such H ∈ Cut(G).
Since every matching generating polynomial is real-rooted, we obtain the following.
Corollary 4.4. Let G be a finite graph with n vertices. If G has no even cycles, then the γpolynomial of the h * -polynomial of A G is g(G, 2x). In particular, h * (A G , x) is real-rooted. Moreover, the normalized volume of A G is 2 n g(G, 1/2).
An example of the suspension of a cactus graph is a wheel graph. In [8], the normalized volume of the symmetric edge polytope of a wheel graph was computed. By using Theorem 1.1, we compute the γ-polynomial of the h * -polynomial and the normalized volume of the polytope.
Example 4.5. Let C n be a cycle of length n. Then C n is a wheel graph. It is known that where L n (x) is the Lucas polynomial defined by See, e.g., [31, p.27 In particular, we obtain This coincides with [8,Theorem 4.24].

REAL-ROOTEDNESS OF h * (A G , x)
In this section, we prove Theorem 1.2. The polynomial is strongly related with the µ-polynomial in methods from chemical graph theory. Suppose that G has n vertices and r cycles C 1 , . . . ,C r . Let t = (t 1 , . . . ,t r ) ∈ R r be a vector whose component t i is associated to the cycle C i for i = 1, 2, . . ., r. It is known [14, Proposition 1a] that the µ-polynomial µ(G, t, x) of a graph G satisfies where R 2 (G) is the set of all subgraphs of G consisting of vertex-disjoint cycles, c(R) is the number of the cycles of R. This polynomial generalizes important graph polynomials.
In fact, we have µ(G, 0, x) = α(G, x) and µ(G, 1, x) = ϕ(G, x), where ϕ(G, x) is the characteristic polynomial of G. See [13,Theorem 5.3.3]. For a cactus graph, the following is known. Proof of Theorem 1.2. It is enough to show that γ(G, x) is real-rooted. The polynomial γ(G, x) satisfies where t = (t 1 , . . .,t r ) with if C i is an even cycle, 0 otherwise.
By Proposition 5.1, this is real-rooted. If u is a root of γ(G, x), then u = 0 and x = 1/ √ −2u is a root of µ(G, t, x). Since 1/ √ −2u is real, so is u.
From Theorem 1.2 the h * -polynomial of the symmetric edge polytope of a wheel graph, i.e., the suspension of a cycle is real-rooted. However, the h * -polynomial of the symmetric edge polytope of a cycle is not always real-rooted.
An n-dimensional simplicial complex ∆ is said to be • flag if all minimal non-faces of ∆ contain only two elements; • balanced if there is a proper coloring of its vertices c : V → [n + 1], where V is the vertex set of ∆.
Frohmader showed that the f -vector of a flag simplicial complex coincides with that of a balanced simplicial complex. Nevo and Petersen posed the following strengthening problem of Conjecture 2.4.
In this section, we discuss this problem for h * (A G , x). In particular, we prove Theorem 1.3. First, we recall that every flag simplicial complex arises from a finite simple graph. Let G be a finite simple graph on [n] with the edge set E(G). A subset C of [n] is called a clique of G if for all i and j belonging to C with i = j, one has {i, j} ∈ E(G). The clique complex of G is the simplicial complex ∆(G) on [n] whose faces are the cliques of G. Proof. It follows that the matching generating polynomial g(G, x) of G and the independence polynomial i(L(G), x) of the line graph L(G) of G are identical. Namely, one has g(G, x) = i(L(G), x). Hence g(G, x) is the f -polynomial of a flag simplicial complex that is the clique complex of L(G).
Moreover, by using the above correspondence, we prove the following proposition.  Proof of Proposition 6.4. Let ∆ be a flag simplicial complex whose f -polynomial is f (x). Then there exists a graph G such that ∆ is the clique complex of G. Moreover, f (x) is the independence polynomial i(G, x) of G. We consider the lexicographic product G[K m ] of G and a complete graph K m of m vertices. It then follows from Lemma 6.5 that Example 6.6. Let C n be an odd cycle of length n. It then follows from Example 4.5 that In the rest of this section, we show that for any bipartite graph G such that every cycle of length ≥ 6 in G has a chord, ∂ A G has a flag unimodular triangulation. In particular, for any forest G, ∂ A G has a flag unimodular triangulation. First, we introduce the theory of Gröbner bases of toric ideals. (See, e.g., [16,Chapter 3] for details on toric ideals and Gröbner bases.) Let P ⊂ R n be a lattice polytope, where P ∩ Z n = {a 1 , . . ., a m }. For simplicity, we assume that P is spanning, i.e., Z n+1 = ∑ m i=1 Z(a i , 1). Note that for any graph G, A G is unimodularly equivalent to a full-dimensional lattice polytope satisfying this condition. Let R = K[t 1 ,t −1 1 , . . .,t n ,t −1 n , s] be the Laurent polynomial ring over a field K and let be the polynomial ring over K. We define the ring homomorphism π : S → R by setting π(x i ) = t a i1 1 · · ·t a in n s where a i = (a i1 , . . . , a in ). The toric ideal I P of P is the kernel of π. It is known that I P is generated by homogeneous binomials. Given a monomial order <, the initial ideal in < (I P ) of I P with respect to < is an ideal generated by the initial 13 monomials in < ( f ) of nonzero polynomials f in I P . The initial complex ∆(P, <) of P with respect to < is where in < (I P ) is the radical of in < (I P ).
We have the following proposition from a fact [32, Proposition 8.6] on the initial complex with respect to a reverse lexicographic order. Proposition 6.8. Suppose that a 1 = 0 is the unique lattice point in the interior of P. Let < be a reverse lexicographic order such that the smallest variable is x 1 . Then 0 is a vertex of every maximal simplex in ∆(P, <), and is a triangulation of the boundary ∂ P of P. In particular, if ∆(P, <) is flag and unimodular, then so is ∆. be the polynomial ring over K. We define the ring homomorphism π : S → R by setting π(z) = s, π(x k ) = t i t −1 j s and π(y k ) = t −1 i t j s if e k = {i, j} ∈ E(G) and i < j. Then the toric ideal I A G of A G is the kernel of π. The initial ideal of I A G plays an important role in, e.g., [8,18,25,27]. In particular, a Gröbner basis of I A G of a graph G with respect to a certain reverse lexicographic order is given in [18,Proposition 3.8].
Theorem 6.9. Let G be a bipartite graph such that every cycle of length ≥ 6 in G has a chord. Then there exists a reverse lexicographic order < such that (i) z is the smallest variable with respect to <; (ii) in < (I A G ) is generated by squarefree quadratic monomials.
In particular, ∂ A G has a flag unimodular triangulation.
Proof. Let [n] be the vertex set of G. Recall that G is a bipartite graph on [n + 2] whose edge set is Since every cycle of length ≥ 6 in G has a chord, every cycle of length ≥ 6 in G has a chord. From [24,Theorem 4.4], there exists a reverse lexicographic order < such that z is the smallest variable with respect to <, and that the initial ideal of I A G with respect to < is generated by squarefree quadratic monomials {m 1 , . . . , m s }. Note that G is obtained from G by contracting the edge {n + 1, n + 2}, and there is a natural correspondence between E( G) \ {{n + 1, n + 2}} and E( G). From a fact shown in the proof of [27,Proposition 5.4], the initial ideal of I A G with respect to the reverse lexicographic order induced by < is generated by squarefree quadratic monomials {m 1 , . . . , m s } \ {x k y k } where e k = {n + 1, n + 2}. Thus ∂ A G has a flag unimodular triangulation by Propositions 6.7 and 6.8.

SYMMETRIC EDGE POLYTOPES OF TYPE B
In this section, we consider the symmetric edge polytope of type B of a cactus bipartite graph. Note that the symmetric edge polytope B G of a graph G is reflexive if and only if G is bipartite [26, Theorem 0.1]. In the case, a formula of the γ-polynomial of h * (B G , x) in terms of interior polynomials is given. Similarly to Theorem 1.1, for a cactus bipartite graph G, we give a formula of γ(B G , x) in terms of matching generating polynomials and show that h * (B G , x) is real-rooted. where t = (t 1 , . . .,t r ) with t i = (−1) |E(C i )| 2 /2. By Proposition 5.1, this is real-rooted. Hence h * (B G , x) is also real-rooted.
By using Theorem 7.2, for an even cycle, we compute the γ-polynomial of the h *polynomial and the normalized volume of the symmetric edge polytope of type B. In particular, we obtain Vol(B C n ) = 2 n γ(B C n , 1/4) = (1 + √ 5) n + (1 − √ 5) n − 2 n .
Theorem 7.2 generalizes the following result. In particular, h * (B G , x) is real-rooted.
Finally, we show that for a forest G, γ(B G , x) coincides with the f -polynomial of a flag simplicial complex. Namely, Nevo-Petersen's conjecture holds for any flag unimodular triangulation of the boundary ∂ B G in this case. Proof. It follows from Corollary 7.4 that the γ-polynomial of h * (B G , x) is g(G, 4x). Thus Propositions 6.3 and 6.4 guarantee that g(G, 4x) is the f -polynomial of a flag simplicial complex.
Remark 7.6. It follows from the proof of [26, Theorem 2.6] that for any forest G, ∂ B G has a flag unimodular triangulation.