Reflexive polytopes arising from bipartite graphs with $\gamma$-positivity associated to interior polynomials

In this paper, we introduce polytopes ${\mathcal B}_G$ arising from root systems $B_n$ and finite graphs $G$, and study their combinatorial and algebraic properties. In particular, it is shown that ${\mathcal B}_G$ is a reflexive polytope with a regular unimodular triangulation if and only if $G$ is bipartite. This implies that the $h^*$-polynomial of ${\mathcal B}_G$ is palindromic and unimodal when $G$ is bipartite. Furthermore, we discuss stronger properties, the $\gamma$-positivity and the real-rootedness of the $h^*$-polynomials. In fact, if $G$ is bipartite, then the $h^*$-polynomial of ${\mathcal B}_G$ is $\gamma$-positive and its $\gamma$-polynomial is given by an interior polynomial (a version of Tutte polynomial of a hypergraph). The $h^*$-polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted. From a counterexample of Neggers--Stanley conjecture, we give a bipartite graph $G$ whose $h^*$-polynomial is not real-rooted but $\gamma$-positive, and coincides with the $h$-polynomial of a flag triangulation of a sphere.


INTRODUCTION
Ardila et al. [1] constructed a unimodular triangulation of the convex hull of the roots of the classical root lattices of type A n , B n , C n and D n , and gave an alternative proof for the known growth series of these root lattices by using the triangulation. On the other hand, polytopes of the root system of type A n arising from finite graphs are called symmetric edge polytopes and their combinatorial properties are well-studied ( [20,21,28]). In this paper, we introduce polytopes arising from the root system of type B n and finite graphs, and study their algebraic and combinatorial properties.
A lattice polytope P ⊂ R d is a convex polytope all of whose vertices have integer coordinates. Let G be a finite simple undirected graph on the vertex set [d] = {1, . . ., d} with the edge set E(G). Let B G ⊂ R d denote the convex hull of the set where e i is i-th unit coordinate vector in R d . Then dim B G = d and B G is centrally symmetric, i.e., for any facet F of B G , −F is also a facet of B G , and the origin 0 of R d is the unique interior lattice point of B G . Note that, if G is a complete graph, then B G coincides with the convex hull of the roots of the root lattices of type B n studied in [1]. Several classes of lattice polytopes arising from graphs have been studied from viewpoints of combinatorics, graph theory, geometric and commutative algebra. In particular, edge polytopes give interesting examples in commutative algebra ( [16,31,32,33,39]). Note that edge polytopes of bipartite graphs are called root polytopes and play important roles in the study of generalized permutohedra ( [37]) and interior polynomials ( [23]).
There is a good relation between B G and edge polytopes. In fact, one of the key properties of B G is that B G is divided into 2 d edge polytopes of certain non-simple graphs G (Proposition 1.1). This fact helps us to find and show interesting properties of B G . In Section 1, by using this fact, we will classify graphs G such that B G has a unimodular covering (Theorem 1.3).
On the other hand, the fact that B G has a unique interior lattice point 0 leads us to consider when B G is reflexive. A lattice polytope P ⊂ R d of dimension d is called reflexive if the origin of R d is a unique lattice point belonging to the interior of P and its dual polytope P ∨ := {y ∈ R d : x, y ≤ 1 for all x ∈ P} is also a lattice polytope, where x, y is the usual inner product of R d . It is known that reflexive polytopes correspond to Gorenstein toric Fano varieties, and they are related to mirror symmetry (see, e.g., [2,8]). In each dimension there exist only finitely many reflexive polytopes up to unimodular equivalence ( [26]) and all of them are known up to dimension 4 ( [25]). In Section 2, we will classify graphs G such that B G is a reflexive polytope. In fact, we will show the following.
Theorem 0.1. Let G be a finite graph. Then the following conditions are equivalent: (i) B G is reflexive and has a regular unimodular triangulation; Moreover, by characterizing when the toric ideal of B G has a Gröbner basis consisting of quadratic binomials for a bipartite graph G, we can classify graphs G such that B G is a reflexive polytope with a flag regular unimodular triangulation. In fact, Theorem 0.2. Let G be a bipartite graph. Then the following conditions are equivalent: (i) The reflexive polytope B G has a flag regular unimodular triangulation; (ii) Any cycle of G of length ≥ 6 has a chord ("chordal bipartite graph"). Now, we turn to the discussion of the h * -polynomial h * (B G , x) of B G . Thanks to the key property (Proposition 1.1), we can compute the h * -polynomial of B G in terms of that of edge polytopes of some graphs. On the other hand, since it is known that the h *polynomial of a reflexive polytope with a regular unimodular triangulation is palindromic and unimodal ( [6]), Theorem 1.3 implies that the h * -polynomial of B G is palindromic and unimodal. In Section 3, we will show a stronger result, which is for any bipartite graph G, the h * -polynomial h * (B G , x) is γ-positive. The theory of interior polynomials (a version of Tutte polynomials of hypergraphs) introduced by Kálmán [22] and the theory of generalized permutohedra [29,37] where G is a connected bipartite graph defined in (1) later and I G (x) is the interior polynomial of G. In particular, h * (B G , x) is γ-positive. Moreover, h * (B G , x) is realrooted if and only if I G (x) is real-rooted.
In addition, we discuss the relations between interior polynomials and other important polynomials in combinatorics.
• If G is bipartite, then the interior polynomial of G is described in terms of kmatching of G (Proposition 3.3); • If G is a forest, then the interior polynomial of G coincides with the matching generating polynomial of G (Proposition 3.4); • If G is a bipartite permutation graph associated with a poset P, then the interior polynomial of G coincides with the P-Eulerian polynomial of P (Proposition 3.5). By using these results and a poset appearing in [43] as a counterexample of Neggers-Stanley conjecture, we will give an example of a centrally symmetric reflexive polytope such that the h * -polynomial is γ-positive and not real-rooted (Example 3.6). This h *polynomial coincides with the h-polynomial of a flag triangulation of a sphere (Proposition 3.7). Hence this example is a counterexample of "Real Root Conjecture" that has been already disproved by Gal [10]. Finally, inspired by the simple description for the h * -polynomials of symmetric edge polytopes of complete bipartite graphs [21], we will compute the h * -polynomial of B G when G is a complete bipartite graph (Example 3.8).
Acknowledgment. The authors were partially supported by JSPS KAKENHI 18H01134 and 16J01549.

A KEY PROPERTY OF B G AND UNIMODULAR COVERINGS
In this section, we see a relation between B G and edge polytopes. First, we recall what edge polytopes are. Let G be a graph on [d] (only here we do not assume that G has no loops) with the edge (including loop) set E(G). Then the edge polytope P G of G is the convex hull of {e i + e j : {i, j} ∈ E(G)}. Note that P G is a (0, 1)-polytope if and only if G has no loops. Given a graph G on [d], let G be a graph on [d + 1] whose edge set is Here, {d + 1, d + 1} is a loop (a cycle of length 1) at d + 1. If G is a bipartite graph with a bipartition V 1 ∪V 2 = [d], let G be a connected bipartite graph on [d + 2] whose edge set is Now, we show the key proposition of this paper. Given ε = (ε 1 , . . . , Proposition 1.1. Work with the same notation as above. Then we have the following: (a) Each B G ∩ O ε is the convex hull of the set B(G) ∩ O ε and unimodularly equivalent to the edge polytope P G of G. Moreover, if G is bipartite, then B G ∩ O ε is unimodularly equivalent to the edge polytope P G of G. In particular, one has Vol(B G ) = 2 d Vol(P G ).  3 and each a i belongs to B(G). Suppose that k-th component of a i is positive and k-th component of a j is negative. Then a i and a j satisfy one of the following conditions: a j = −a i , and a i + a j = 0 + 0, a i = e k , a j = −e k ± e k ′ , and a i + a j = ±e k ′ + 0, a i = e k ± e k ′ , a j = −e k , and a i + a j = ±e k ′ + 0, a i = e k ± e k ′ , a j = −e k ± e k ′′ ( = −a i ), and a i + a j = ±e k ′ ± e k ′′ ( = 0). By using the above relations for a i + a j finitely many times, we may assume that k-th component of each vector a i is nonnegative (resp. nonpositive) if x k ≥ 0 (resp. x k ≤ 0). Then each a i belongs to B(G) ∩ O ε and hence x ∈ P.
Next, we show that each B G ∩ O ε is unimodularly equivalent to the edge polytope P G . 1) . It is easy to see that each B G ∩ O ε is unimodularly equivalent to Q for all ε. Moreover, one has Let P ⊂ R d be a lattice polytope of dimension d. We now focus on the following properties.
(VA) We say that P is very ample if for all sufficiently large k ∈ Z ≥1 and for all x ∈ kP ∩ Z d , there exist x 1 , . . ., x k ∈ P ∩ Z d with x = x 1 + · · · + x k . (IDP) We say that P possesses the integer decomposition property (or is IDP for short) if for all k ∈ Z ≥1 and for all x ∈ kP ∩ Z d , there exist x 1 , . . . , x k ∈ P ∩ Z d with x = x 1 + · · · + x k . (UC) We say that P has a unimodular covering if there exist unimodular lattice simplices ∆ 1 , . . . , ∆ n such that P = 1≤i≤n ∆ i . (UT) We say that P has a unimodular triangulation if P admits a lattice triangulation consisting of unimodular lattice simplices. These properties satisfy the implications On the other hand, it is known that the opposite implications are false. However, for edge polytopes, the first three properties are equivalent. We say that a graph G satisfies the odd cycle condition if, for any two cycles C 1 and C 2 that belong to the same connected component of G and have no common vertices, there exists an edge {i, j} of G such that i is a vertex of C 1 and j is a vertex of C 2 . The following fact is known ( [7,31,39]). Proposition 1.2. Let G be a finite (not necessarily simple) graph. Suppose that there exists an edge {i, j} of G if G has loops at i and j with i = j. Then the following conditions are equivalent: (i) P G has a unimodular covering; (ii) P G is IDP; 4 (iii) P G is very ample; (iv) G satisfies the odd cycle condition.
We now show that the same assertion holds for B G . Namely, we prove the following. Theorem 1.3. Let G be a finite simple graph. Then the following conditions are equivalent: Suppose that G does not satisfy the odd cycle condition. By Proposition 1.2, the edge polytope P G of G is not very ample. Since P G is a face of B G by Proposition 1.1 (b), B G is not very ample.
(iv) ⇒ (i): Suppose that G satisfies the odd cycle condition. Then so does G. Hence Proposition 1.2 guarantees that P G has a unimodular covering. By Proposition 1.1 (a), B G has a unimodular covering. Figure 1. Since G satisfies the odd cycle condition, FIGURE 1. A graph in [32] B G has a unimodular covering. However, since the edge polytope P G has no regular unimodular triangulations ( [32]), so does B G by Proposition 1.1 (b). We do not know whether B G has a (nonregular) unimodular triangulation or not.

REFLEXIVE POLYTOPES AND FLAG TRIANGULATIONS OF B G
In the present section, we classify graphs G such that • B G is a reflexive polytope.
• B G is a reflexive polytope with a flag regular unimodular triangulation. Namely, we prove Theorems 0.1 and 0.2. First, we see some examples that B G is reflexive.
Examples 2.1. (a) If G is an empty graph, then B G is a cross polytope.
(b) Let G be a complete graph with 2 vertices. Then B G ∩ Z 2 is the column vectors of the matrix and B G is reflexive and has a regular unimodular triangulation. Since the matrix is not unimodular, we cannot apply [34, Lemma 2.11] to show this fact.
In order to show that a lattice polytope is reflexive, we use an algebraic technique on Gröbner bases. We recall basic materials and notation on toric ideals. Let Let P ⊂ R d be a lattice polytope and P ∩ Z d = {a 1 , . . . , a n }. Then, the toric ring of P is the subalgebra K[P] of K[t ±1 , s] generated by {t a 1 s, . . ., t a n s} over K. We regard K[P] as a homogeneous algebra by setting each deg t a i s = 1. Let K[x] = K[x 1 , . . ., x n ] denote the polynomial ring in n variables over K. The toric ideal I P of P is the kernel of the surjective homomorphism π : It is known that I P is generated by homogeneous binomials. See, e.g., [44]. Let < be a monomial order on K[x] and in < (I P ) the initial ideal of I P with respect to <. The initial ideal in < (I P ) is called squarefree (resp. quadratic) if in < (I P ) is generated by squarefree (resp. quadratic) monomials. Now, we introduce an algebraic technique to show that a lattice polytope is reflexive.
. Let P ⊂ R d be a lattice polytope of dimension d such that the origin of R d is contained in its interior and P ∩ Z d = {a 1 , . . . , a n }. Suppose that any lattice point in Z d+1 is a linear integer combination of the lattice points in P × {1} and there exists an ordering of the variables x i 1 < · · · < x i n for which a i 1 = 0 such that the initial ideal in < (I P ) of I P with respect to the reverse lexicographic order < on K[x] induced by the ordering is squarefree. Then P is reflexive and has a regular unimodular triangulation.
By using this technique, several families of reflexive polytopes with regular unimodular triangulations are given in [13,14,15,17,18,19,35]. In order to apply Lemma 2.2 to show Theorem 0.1, we see a relation between the toric ideal of B G and that of P G . Let G be a simple graph on [d] with edge set E(G) and let R G denote the polynomial ring in d + 1 + |E(G)| variables over K. Then the toric ideal I P G of P G is the kernel of π| S G . For each ε = (ε 1 , . . . , ε d ) ∈ {−1, 1} d , we define a ring homomorphism ϕ ε : S G → R G by ϕ ε (x i+ ) = x iα and ϕ ε (y i j++ ) = y i jαβ where α is the sign of ε i and β is the sign of ε j . In particular, ϕ (1,...,1) : S G → R G is an inclusion map. Lemma 2.3. Let G be a Gröbner basis of I P G with respect to a reverse lexicographic order < S on S G such that z < {x i+ } < {y i j++ }. Let < R be a reverse lexicographic order such is a Gröbner basis of I B G with respect to < R , where the underlined monomial is the initial monomial of each binomial. (Here we identify y i jαβ with y jiβ α .) In particular, if in < S (I P G ) is squarefree (resp. quadratic), then so is in < R (I B G ).
Proof. It is easy to see that G ′ is a subset of I B G . Assume that G ′ is not a Gröbner basis of I B G with respect to < R . Let in(G ′ ) = in < R (g) : g ∈ G ′ . Then there exists a non-zero irreducible homogeneous binomial f = u − v ∈ I B G such that neither u nor v belongs to in(G ′ ). Since both u and v are divided by none of x i+ x i− , x iα y i jβ γ , y i jαγ y ikβ δ (α = β ), they are of the form Since f belongs to I B G , the exponent of t ℓ in π(u) and π(v) are the same. Hence, one of x ℓα ℓ and y ℓmα ℓ α m appears in u if and only if one of x ℓα ′ ℓ and y ℓnα ′ ℓ α ′ n appears in v with α ℓ = α ′ ℓ . Let ε be a vector in {−1, 1} d such that the sign of the ith component of ε is α i if one of x iα i and y i jα i α j appears in u. Then f belongs to the ideal ϕ ε (I P G ). Let f ′ ∈ I P G be a binomial such that ϕ ε ( f ′ ) = f . Since G is a Gröbner basis of I P G , there exists a binomial g ∈ G whose initial monomial in < R (g) divides the one of the monomials in f ′ . By the definition of < R , we have in < R (ϕ ε (g)) = ϕ ε (in < R (g)). Hence in < R (ϕ ε (g)) divides one of the monomials in f = ϕ ε ( f ′ ). This is a contradiction.
Using this Gröbner basis with respect to a reverse lexicographic order, we verify which B G is a reflexive polytope. Namely, we prove Theorem 0.1.
(ii) ⇒ (iii): Suppose that G is not bipartite. Let G 1 , . . ., G s be connected components of G and let d i be the number of vertices of G i . In particular, is a non-bipartite graph on the vertex set {p 1 , . . . , p ℓ }, and w i = ∑ ℓ k=1 2e p k ∈ R d if G i is a bipartite graph whose vertices are divided into two independent sets {p 1 , . . . , p ℓ } and {q 1 , . . . , q m }. It then follows that H = {x ∈ R d : w · x = 2} is a supporting hyperplane of B G and the corresponding face If G i is not bipartite, then the dimension of conv( Suppose that G is bipartite. Let < S and < R be any reverse lexicographic orders satisfying the condition in Lemma 2.3. It is well-known that any triangulation of the edge polytope of a bipartite graph is unimodular. By [44,Corollary 8.9], the initial ideal of the toric ideal of P G with respect to < S is squarefree. Thanks to Lemmas 2.2 and 2.3, we have a desired conclusion. We now give a theorem on quadratic Gröbner bases of I B G when G is bipartite. This theorem implies that Theorem 0.2. The same result is known for edge polytopes ( [33]).
Theorem 2.4. Let G be a bipartite graph. Then the following conditions are equivalent: (i) The toric ideal I B G of B G has a squarefree quadratic initial ideal (i.e., B G has a flag regular unimodular triangulation); (ii) The toric ring K[B G ] of B G is a Koszul algebra; (iii) The toric ideal I B G of B G is generated by quadratic binomials; (iv) Any cycle of G of length ≥ 6 has a chord ("chordal bipartite graph").
(iii) ⇒ (iv): Suppose that G has a cycle of length ≥ 6 without chords. By the theorem in [33], the toric ideal of P G is not generated by quadratic binomials. Since the edge polytope P G is a face of B G , the toric ring K[P G ] is a combinatorial pure subring [30] of K[B G ]. Hence I B G is not generated by quadratic binomials.
(iv) ⇒ (i): Suppose that any cycle of G of length ≥ 6 has a chord. By Lemma 2.3, it is enough to show that the initial ideal of I P G is squarefree and quadratic with respect to a reverse lexicographic order < S such that z < {x i+ } < {y kℓ++ }. Let A = (a i j ) be the incidence matrix of G whose rows are indexed by V 1 and whose columns are indexed by V 2 . Then the incidence matrix of G is By the same argument as in the proof of [33], we may assume that A ′ contains no submatrices 1 1 1 0 if we permute the rows and columns of A in A ′ . Each quadratic binomial in I P G corresponds to a submatrix 1 1 1 1 of A ′ . The proof of the theorem in [33] 8 guarantees that the initial ideal is squarefree and quadratic if the initial monomial of each quadratic binomial corresponds to 1 1 . It is easy to see that there exists a such reverse lexicographic order which satisfies z < {x i+ } < {y kℓ++ }.

γ-POSITIVITY AND REAL-ROOTEDNESS OF THE h * -POLYNOMIAL OF B G
In this section, we study the h * -polynomial of B G for a graph G. First, we recall what h * -polynomials are. Let P ⊂ R d be a lattice polytope of dimension d. Given a positive integer n, we define The study on L P (n) originated in Ehrhart [9] who proved that L P (n) is a polynomial in n of degree d with the constant term 1. We say that L P (n) is 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 well known that it can be expressed as a rational function of the form The polynomial h * (P, x) is a polynomial in x of degree at most d with nonnegative integer coefficients ( [40]) and it is called the h * -polynomial (or the δ -polynomial) of P. Moreover, one has Vol(P) = h * (P, 1), where Vol(P) is the normalized volume of P. Thanks to Proposition 1.1 (a), we give a formula for h * -polynomial of B G in terms of that of edge polytopes of some graphs. By the following formula, we can calculate the h * -polynomial of B G if we can calculate each h * (P H , x).
where S j (G) denote the set of all induced subgraph of G with j vertices.
Proof. By Proposition 1.1 (a), B G is divided into 2 d lattice polytopes of the form B G ∩ O ε . Each B G ∩ O ε is unimodularly equivalent to P G . In addition, the intersection of where G ′ is the induced subgraph of G obtained by deleting the vertex k, and ε ′′ is obtained by deleting k-th component of ε. Hence the Ehrhart polynomial L B G (n) satisfies the following: L P H (n). 9 Thus the Ehrhart series satisfies as desired.
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.
(LC) We say that f is log-concave if a 2 i ≥ a i−1 a i+1 for all i. (UN) We say that f is unimodal if a 0 ≤ a 1 ≤ · · · ≤ a k ≥ · · · ≥ a d for some k. If all its coefficients are nonnegative, then these properties satisfy the implications On the other hand, the polynomial f is said to be palindromic if We can see that a γ-positive polynomial is real-rooted if and only if its γ-polynomial had only real roots.
By the following proposition, we are interested in connected bipartite graphs.
Proposition 3.2. Let G be a bipartite graph and G 1 , . . . , G s the connected components of G. Then the h * -polynomial of B G is palindromic, unimodal and Proof. It is known [12] that the h * -polynomial of a lattice polytope P with the interior lattice point 0 is palindromic if and only if P is reflexive. Moreover, if a reflexive polytope P has a unimodular triangulation, then the h * -polynomial of P is unimodal (see [6]). It is easy to see that, B G is the free sum of B G 1 , . . . , B G s . Thus we have a desired conclusion by Theorem 0.1 and [5, Theorem 1].
In the rest of the present paper, we discuss the γ-positivity and the real-rootedness on the h * -polynomial of B G when G is a bipartite graph. The edge polytope of a bipartite graph G is called a root polytope of G and it is shown [23] that h * -vector of P G coincides with interior polynomial I G (x) of a hypergraph induced by G. First, we discuss interior polynomials introduced by Kálmán [22] and developed in many papers.
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 with a bipartition V ∪ E such that {v i , e j } is an edge of BipH if v i ∈ e j . Assume that BipH is connected. A hypertree in H is a function f : E → {0, 1, . . .} 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 induce f. Let B 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 function I H (x) = ∑ f∈B H x ι(f) . It is known [22, Proposition 6.1] that deg I H (x) ≤ min{|V |, |E|} − 1. If G = BipH , then we set I G (x) = I H (x). Kálmán and Postnikov [23] proved that for a connected bipartite graph G. Note that if G is a bipartite graph, then the bipartite graph G arising from G is connected. Hence we can use this formula to study equation (2) in Proposition 3.1. (Interior polynomials of disconnected bipartite graphs are defined in [24].) A k-matching of G is a set of k pairwise non-adjacent edges of G. Let M(G, k) = {v i 1 , . . . , v i k , e j 1 , . . ., e j k } : there exists a k-matching of G whose vertex set is {v i 1 , . . . , v i k , e j 1 , . . ., e j k } .
For k = 0, we set M(G, 0) = { / 0}. Using the theory of generalized permutohedra [29,37], we have the following important fact on interior polynomials:  Given a hypertree f ∈ HT( G), let Γ be a spanning tree that induces f. We now repeat the following procedure for Γ: • For each j = 1, 2, . . ., n, since Γ is a spanning tree, there exists a unique path e j v i · · · v 1 from e j to v 1 . If i > 1, then remove {v i , e j } from Γ and add {v 1 , e j } to Γ. It then follows that the new Γ is a spanning tree that induces f.
Hence we may assume that {v 1 , e j } is an edge of Γ for all 1 ≤ j ≤ n. Note that the degree of each v i (2 ≤ i ≤ m) is 1. By definition, e 1 is always internally active. We show that, e j ( j ≥ 2) is internally active if and only if f(e j ) = 0. By definition, if f(e j ) = 0, then e j is internally active. Suppose f(e j ) > 0. Then there exists i ≥ 2 such that {v i , e j } is an edge of Γ. Let f ′ ∈ HT( G) be a hypertree induced by a spanning tree obtained by replacing {v i , e j } with {v i , e 1 } in Γ.
Then we have f ′ (e j ) = f(e j ) − 1, f ′ (e 1 ) = f(e 1 ) + 1 and f ′ (e k ) = f(e k ) for all 1 < k = j. Hence e j is not internally active. Thus ι(f) is the number of e j ( j ≥ 2) such that there exists an edge {v i , e j } of Γ f for some i ≥ 2.
In order to prove the equation (3), it is enough to show that, for fixed hyperedges e j 1 , . . ., e j k with 2 ≤ j 1 < · · · < j k ≤ n, the cardinality of there exists a k-matching of G whose vertex set is {v i 1 , . . ., v i k , e j 1 , . . . , e j k } .

11
Let G j 1 ,..., j k be the induced subgraph of G on the vertex set V ∪ {e j 1 , . . . , e j k }. If e j ℓ is an isolated vertex in G j 1 ,..., j k , then both S j 1 ,..., j k and M j 1 ,..., j k are empty sets. If v i is an isolated vertex in G j 1 ,..., j k , then there is no relations between v i and two sets, and hence we can ignore v i . Thus we may assume that G j 1 ,..., j k has no isolated vertices. It is known that M j 1 ,..., j k is the set of bases of a transversal matroid associated with G j 1 ,..., j k . See, e.g., [36]. For i = 2, . . ., m, let Oh [29] define a lattice polytope P M j 1 ,..., j k to be the generalized permutohedron [37] of the induced subgraph of G on the vertex set V ∪ {e 1 , e j 1 , . . . , e j k }, i.e., P M j 1 ,..., j k is the Minkowski sum ∆ I 2 + · · · + ∆ I m , where ∆ I = Conv({e j : j ∈ I}) ⊂ R k+1 and e 0 , e 1 , . . . , e k are unit coordinate vectors in R k+1 . By [29,Lemma 22 and Proposition 26], the cardinality of M j 1 ,..., j k is equal to the number of the lattice point (x 0 , x 1 , . . ., x k ) ∈ P M j 1 ,..., j k ∩ Z k+1 with x 1 , x 2 , . . ., x k ≥ 1. In addition, by [37,Proposition 14.12], any lattice point . . , f(e j k )) with f ∈ S j 1 ,..., j k , it follows that the number of the lattice point (x 0 , x 1 , . . . , x k ) ∈ P M j 1 ,..., j k ∩ Z k+1 with x 1 , x 2 , . . . , x k ≥ 1 is equal to the cardinality of S j 1 ,..., j k , as desired. Now, we show that the h * -polynomial of B G is γ-positive if G is a bipartite graph. In fact, we prove Theorem 0.3.

Thus we have
By Proposition 3.3, it follows that, if G is a forest, then I G (x) coincides with the matching generating polynomial of G.
Proposition 3.4. Let G be a forest. Then we have where m k (G) is the number of k-matching in G. In particular, I G (x) is real-rooted.
Proof. Let M 1 and M 2 be k-matchings of G. Suppose that M 1 and M 2 have the same vertex set {v i 1 , . . . , v i k , e j 1 , . . . , e j k }.
is not empty, then M corresponds to a subgraph of G such that the degree of each vertex is 2. Hence M has at least one cycle. This contradicts that G is a forest. Hence we have M 1 = M 2 . Thus m k (G) is the cardinality of M(G, k). In general, it is known that ∑ k≥0 m k (G) x k is real-rooted for any graph G. See, e.g., [11,27].
Next we will show that, if a bipartite graph G is a "permutation graph" associated with a poset P, then the interior polynomial I G (x) coincides with P-Eulerian polynomial W (P)(x). A permutation graph is a graph on [d] with edge set {{i, j} : L i and L j intersect each other}, where there are d points 1, 2, . . ., d on two parallel lines L 1 and L 2 in the plane, and the straight lines L i connect i on L 1 and i on L 2 . If G is a bipartite graph with a bipartition V 1 ∪V 2 , the following conditions are equivalent : (i) G is permutation; (ii) The complement of G is a comparability graph of a poset; (iii) There exist orderings < 1 on V 1 and < 2 on V 2 such that i, i ′ ∈ V 1 , i < 1 i ′ , j, j ′ ∈ V 2 , j < 2 j ′ , {i, j}, {i ′ , j ′ } ∈ E(G) =⇒ {i, j ′ }, {i ′ , j} ∈ E(G); (iv) For any three vertices, there exists a pair of them such that there exists no path containing the two vertices that avoids the neighborhood of the remaining vertex. See [4] for details. On the other hand, let P be a naturally labeled poset P on [d]. Then the order polynomial Ω(P, m) of P is defined for 0 < m ∈ Z to be the number of orderpreserving maps σ : P → [m]. It is known that where L (P) is the set of linear extensions of P and d(π) is the number of descent of π. The P-Eulerian polynomial W (P)(x) is defied by W (P)(x) = ∑ π∈L (P) x d(π) .
See, e.g., [42] for details. We now give a relation between the interior polynomial and the P-Eulerian polynomial of a finite poset.
Proposition 3.5. Let G be a bipartite permutation graph and let P be a poset whose comparability graph is the complement of a bipartite graph G. Then we have I G (x) = W (P)(x).
Proof. In this case, B G ∩ O (1,...,1) is the chain polytope C P of P. It is known that the h *polynomial of C P is the P-Eulerian polynomial W (P)(x). See [41,42] for details. Thus we have I G (x) = h * (P G , x) = W (P)(x), as desired.
It was conjectured by Neggers-Stanley that W (P)(x) is real-rooted. However this is false in general. The first counterexample was given in [3] (not naturally labeled posets). Counterexamples of naturally labeled posets were given in [43]. Counterexamples in these two papers are narrow posets, i.e., elements of posets are partitioned into two chains. It is easy to see that P is narrow poset if and only if the comparability graph of P is the complement of a bipartite graph. Since Stembridge found many counterexamples which are naturally labeled narrow posets, there are many bipartite permutation graphs G such that h * (B G , x) are not real-rooted. We give one of them as follows.