Pseudo-Gorenstein edge rings and a new family of almost Gorenstein edge rings
Yuta Hatasa Nobukazu Kowaki Koji Matsushita





[Y. Hatasa]
    Department of Mathematics,
    Tokyo Institute of Technology, 2–12–1 Ōokayama, Meguro-ku,
    Tokyo 152–8551, Japan

[Y. Hatasa]
    International Institute for Sustainability with Knotted Chiral Meta Matter (WPI-SKCM^2),
    Hiroshima University, 1–3–1 Kagamiyama, Higashi-Hiroshima, Hiroshima 739–8526, Japan

hatasa.y.aa@m.titech.ac.jp

[N. Kowaki]Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan
u793177f@ecs.osaka-u.ac.jp

[K. Matsushita]Department of Pure and Applied Mathematics, Graduate School of Information Science and Technology, Osaka University, Suita, Osaka 565-0871, Japan
k-matsushita@ist.osaka-u.ac.jp

[2020]
    Primary
    13D40; 
    Secondary
    13F55, 
    13F65, 
    05C25, 



§ ABSTRACT 
    In this paper, we study edge rings and their h-polynomials.
    We investigate when edge rings are pseudo-Gorenstein, which means that the leading coefficients of the h-polynomials of edge rings are equal to 1.
    Moreover, we compute the h-polynomials of a special family of edge rings and show that some of them are almost Gorenstein.


Pseudo-Gorenstein edge rings and a new family of almost Gorenstein edge rings
    Yuta Hatasa, Nobukazu Kowaki, Koji Matsushita
    September 9, 2024
=============================================================================





§ INTRODUCTION




 §.§ Backgrounds


Let R = ⊕_k≥ 0 R_k be a Cohen–Macaulay homogeneous domain of dimension d over an algebraically closed field R_0 = with characteristic 0.
Then the Hilbert series of R is defined as the formal power series ∑_k≥ 0(_ R_k)t^k, and it is known that we can write the Hilbert series of R as the following form:

    ∑_k ≥ 0 (_ R_k) t^k=h_0+h_1t+⋯+h_st^s/(1-t)^d,

where h_s ≠ 0.
We call the polynomial h_0+h_1t+⋯+h_st^s the h-polynomial of R, denoted by h(R;t), and call the sequence (h_0,h_1,…,h_s) the h-vector of R.
In addition, we also call the index s the socle degree of R, and denoted by s(R).

The h-polynomial (h-vector) of R is one of the most important invariants in the theory of commutative algebra because it tells us what commutative-algebraic properties R has.
In fact, it is well known that R is Gorenstein if and only if R has the symmetric h-vector, that is, h_i=h_s(R)-i holds for each i=0,…,s(R)/2 (<cit.>).
Moreover, recent studies have shown that the h-vector has connections not only to Gorensteinness, but also to its generalized notions such as levelness (<cit.>), almost Gorensteinness (<cit.>) and nearly Gorensteinness (<cit.>).
For example, the following facts are known:

    
  * R is level if and only if the Cohen–Macaulay type of R is equal to h_s(R) (see <cit.>). This implies that if R is level, then h_s(R)=1 if and only if R is Gorenstein;
    
  * If s(R)≥ 2 and R is almost Gorenstein, then h_s(R)=1 (<cit.>);
    
  * If R is nearly Gorenstein, then h_s(R)=1 if and only if R is Gorenstein (<cit.>).

These show that the leading coefficient h_s(R) of the h-polynomial of R has wealth information.
In particular, these motivate determining when h_s(R)=1.
A Cohen–Macaulay homogeneous ring R is called pseudo-Gorenstein if h_s(R)=1 (<cit.>), and certain classes of pseudo-Gorenstein rings are characterized (see, e.g., <cit.>).

In this paper, we study the h-polynomials and pseudo-Gorensteinness of certain homogeneous domains, called edge rings.



For a finite simple graph G on the vertex set V(G)=[d] {1,…,d} with the edge set E(G), we define the edge ring of G as follows:

    [G] [t_i t_j : {i,j}∈ E(G)]⊂[t_1,…,t_d].


Edge rings began to be studied by Ohsugi–Hibi (<cit.>) and Simis–Vasconcelos–Villarreal (<cit.>).
Since then, many researchers have studied the commutative ring-theoretic properties of edge rings.
In particular, the h-polynomials of the edge rings have been investigated.
As far as we know, the h-vectors of the edge rings of the following graphs have been computed:

    
  * Complete graphs and complete bipartite graphs (see <cit.>);
    
  * Bipartite graphs (via interior polynomials) (<cit.>);
    
  * A family of graphs composed of a complete bipartite graph and a cycle graph (<cit.>);
    
  * A family of graphs consisting of odd cycles that share a single common vertex (<cit.>).

Moreover, Gorensteinness, leveleness, almost Gorensteinness, nearly Gorensteinness and pseudo-Gorensteinness of edge rings have been investigated, respectively (see <cit.>).



There are two goals of this paper; one is to characterize when edge rings are pseudo-Gorenstein.
Another one is to find a new family of edge rings having commutative ring-theoretic properties mentioned above and to examine the behavior of their h-polynomials.
In this paper, we focus on almost Gorenstein edge rings and their h-polynomials.





 §.§ Results

First, we completely characterize when the edge rings of bipartite graphs are pseudo-Gorenstein:


    Let G be a bipartite graph.
    Then the edge ring [G] is pseudo-Gorenstein if and only if every block of G is matching-covered.

We also derive some corollaries from <Ref> (<Ref>).

Moreover, we investigate the case of non-bipartite graphs in <Ref>.
This case becomes much more difficult than the bipartite graph case, and the assertions of <Ref> and corollaries mentioned above are no longer true.
However, the assertions can still be ensured with the addition of certain assumptions (<Ref>).



Next, we give a new family of almost Gorenstein edge rings for which their h-vectors are computed.
For integers m, n ≥ 3 and 0 ≤ r ≤min{m, n}, let G_m, n, r be the graph obtained from a
complete bipartite graph K_m, n by removing a matching M with M = r (see <Ref> for the precise definition of G_m,n,r).


    We have
    
    h_[G_m, n, r](t) = 1 +(m - 1)(n - 1) - r t
            + ∑_i = 2^min{m, n}m - 1in - 1i t^i.

    Moreover, [G_m, n, r] is almost Gorenstein if and only if m = n.






 §.§ Organization


The structure of this paper is as follows.
In <Ref>, we prepare the required materials for the discussions later.
We recall definitions and notations associated with commutative algebras and edge rings.
In <Ref>, we discuss the pseudo-Gorensteinness of edge rings.
Before that, we recall some definitions and facts on graph theory.
We then provide the characterization of pseudo-Gorenstein edge rings for bipartite graphs and investigate the case of non-bipartite graphs.
In <Ref>, we introduce a new family of almost Gorenstein edge rings and compute their h-polynomials.
Moreover, we give an observation on the behavior of the h-polynomials of almost Gorenstein edge rings.





 §.§ Acknowledgement

The author would like to thank Akihiro Higashitani, Tamás Kálmán, and Sora Miyashita for their helpful comments and advice on improving this paper.
The third author is partially supported by Grant-in-Aid for JSPS Fellows Grant JP22J20033.



§ PRELIMINARIES




 §.§ Cohen–Macaulay homogeneous domains and their h-polynomials

Throughout this subsection, let R = ⊕_k≥ 0 R_k be a Cohen–Macaulay homogeneous domain of dimension d over an algebraically closed field R_0 = with characteristic 0 and let (h_0,…,h_s) be the h-vector.
First, we introduce some fundamental objects associated with homogeneous rings (consult, e.g., <cit.> for the introduction).

    
  * Let ω_R denote a canonical module of R and let a(R) denote the a-invariant of R, i.e., a(R)=-min{j:(ω_R)_j ≠ 0}.
          When it is clear from context, we simply write a instead of a(R).
    
  * For a graded R-module M, we use the following notation:
          
              
  * Let μ(M) denote the number of minimal generators of M.
              
  * Let e(M) denote the multiplicity of M. Then the inequality μ(M) ≤ e(M) always holds.
              
  * Let M(-ℓ) denote the R-module whose grading is given by M(-ℓ)_k=M_k-ℓ for any k ∈.
          
    
  * As mentioned in the introduction, we denote the h-polynomial (resp. the socle degree) of R by h(R;t) (resp. s(R)).
          We simply write s instead of s(R) as in the a-invariant.
          Note that h_s=_ (ω_R)_-a and d+a=s (see <cit.>).


Next, we recall the definitions and some properties of pseudo-Gorenstein rings and almost Gorenstein rings, respectively.

[<cit.>]
    We call R pseudo-Gorenstein if h_s=_ (ω_R)_-a=1.


[<cit.>]
    We call R almost Gorenstein
    if there exists an exact sequence of graded R-modules
    
    0 → R →ω_R(-a) → C → 0

    with μ(C)=e(C).

Under our assumptions on R, there always exists a degree-preserving injection from R to ω_R(-a) (<cit.>).
Moreover, we see that μ(C)=μ(ω_R)-1 and e(C)=∑_j=0^s-1h_s+⋯+h_s-j-h_0+⋯+h_j (<cit.>), especially, μ(C) and e(C) do not depend on C.

Let e(R) ∑_j=0^s-1h_s+⋯+h_s-j-h_0+⋯+h_j. Note that e(R)≥μ(ω_R)-1.


    Work with the same notation as above.
    Then R is almost Gorenstein if and only if one has e(R)=μ(ω_R)-1.



    If R is almost Gorenstein and s(R) ≥ 2, then h_s(R)=1, that is, R is pseudo-Gorenstein.






 §.§ Edge rings and edge polytopes

Throughout this paper, all graphs are finite and have no loops and no multiple edges.
We recall the definition and some properties of edge rings.
We also recall edge polytopes, which are lattice polytopes arising from graphs.
We can regard edge rings as affine semigroup rings associated with edge polytopes.
See, e.g., <cit.> or <cit.> for the introduction to edge rings.

For a graph G on the vertex set V(G)=[d] with the edge set E(G), we consider the morphism ψ_G of -algebras:

    ψ_G : [x_{i,j} : {i,j}∈ E(G)] →[t_1,…,t_d],  induced by ψ_G(x_{i,j})=t_it_j.

Then we denote the image (resp. the kernel) of ψ_G by [G] (resp. I_G) and call [G] the edge ring of G.
The edge ring [G] is a homogeneous domain by setting (t_it_j)=1 for each i,j∈ E(G).

Given an edge e={i,j}∈ E(G), let ρ(e) _i+_j, where _i denotes the i-th unit vector of ^d for i=1,…,d.
We define the convex polytope associated to G as follows:

    P_G ({ρ(e) : e ∈ E(G)}) ⊂^d.

We call P_G the edge polytope of G.

We can regard [G] as an affine semigroup ring as follows:
Let _G={ρ(e) : e ∈ E(G)} and let S_G be the affine semigroup generated by _G, that is, S_G=_≥ 0_G. Then we can easily see that [G] is isomorphic to the affine semigroup ring of S_G.

Let b(G) be the number of bipartite connected components of G, then we have P_G=d-b(G)-1 (see <cit.>).
This implies that the Krull dimension of [G] is equal to d-b(G).


    Let G be a graph. Then the following are equivalent:
    
        
  * [G] is normal;
        
  * S_G=_≥ 0_G∩_G;
        
  * _[G]_k =kP_G∩^d for any k∈_≥ 0.
        
  * each connected component of G satisfies the odd cycle condition, i.e., for each pair of odd cycles C and C' with no common vertex, there is an edge {v,v'} with v ∈ V(C) and v' ∈ V(C')
    

For a lattice polytope P⊂^d and k∈_≥ 0, L_P(k) kP∩^d is called the Ehrhart polynomial of P.
If [G] is normal, we can compute the Ehrhart polynomial of P_G using the h-vector of [G] as follows:

    
    Let G be a graph with P_G=δ and let (h_0,…,h_s) be the h-vector of [G].
    If [G] is normal, then the Ehrhart polynomial L_P_G(k) of P_G can be described as follows:
    
    L_P_G(k)=h_0k+δδ+h_1k+δ-1δ+⋯ +h_sk+δ-sδ.



Moreover, if [G] is normal, then [G] is Cohen–Macaulay and the ideal generated by monomials corresponding to the elements in ∫(S_G), where ∫(S_G) denotes the set of elements of S_G in the relative interior of the cone generated by S_G, is the canonical module of [G] (see <cit.>).
In particular, let ℓ_G minn : ∫(nP_G)∩^d≠∅, where ∫(P) denotes the relative interior of a polytope P, then we have ℓ_G=-a([G]).

To observe ∫(S_G), we give a description of the cone _≥ 0_G.
To this end, we recall some terminologies and notations.
We say that a subset T of V(G) is an independent set of G if {v,w}∉ E(G) for any v,w∈ T.
Note that each singleton is regarded as an independent set.
We denote the set of non-empty independent sets of G by _G.

For i ∈ [d] and T ∈_G, let

    H_i {(x_1,…,x_d) ∈^d : x_i= 0},
             H_T (x_1,…,x_d) ∈^d : ∑_j ∈ N_G(T)x_j - ∑_i ∈ Tx_i = 0,     
        H_i^+ {(x_1,…,x_d) ∈^d : x_i≥ 0},
             H_T^+ (x_1,…,x_d) ∈^d : ∑_j ∈ N_G(T)x_j - ∑_i ∈ Tx_i ≥ 0, 
        H_i^>{(x_1,…,x_d) ∈^d : x_i > 0},
             H_T^>(x_1,…,x_d) ∈^d : ∑_j ∈ N_G(T)x_j - ∑_i ∈ Tx_i > 0,

where N_G(T) v∈ [d] : v,w∈ E(G)  for some w∈ T.

The following terminologies are used in <cit.>: Suppose that G is connected.

    
  * For a subset W ⊂ V(G), let G∖ W denote the induced subgraph with respect to V(G)∖ W (for a vertex v, we denote by G ∖ v instead of G ∖{v}).
    
  * We call a vertex v of G ordinary if G ∖ v is connected.
    
  * Given an independent set T ⊂ V(G), let B(T) denote the bipartite graph on T ∪ N_G(T) with the edge set {{v,w}∈ E(G) : v ∈ T, w ∈ N_G(T)}.
    
  * When G is a bipartite graph with the partition V(G)=V_1⊔ V_2, a non-empty set T ⊂ V_1 is said to be an acceptable set if the following conditions are satisfied:
          
              
  * B(T) is connected;
              
  * G ∖ V(B(T)) is a connected graph with at least one edge.
          



    Let G be a connected graph.
    Then the cone _≥ 0_G has the following representation:
    
    _≥ 0_G=⋂_i∈ [d]H^+_i∩⋂_T∈_GH^+_T.

    Moreover, if G is bipartite, each facet of P_G is defined by a supporting hyperplane H_i for some ordinary vertex i or H_T for some acceptable set T.


For a subset F⊂ E(G), we set v_F ∑_e∈ Fρ(e).
The following lemma is obvious but important to our results:

    Let G be a graph.
    For F⊂ E(G) and T∈_G, we have v_F ∈ H_T^> if there exists f∈ F with f∈ E(G)∖ E(B(T)) and f∩ N_G(T)≠∅.


A connected graph G is said to be 2-connected if all vertices of G are ordinary.
A block is a maximal 2-connected component of G.


    Let G be a bipartite graph and let B_1,…, B_n be their blocks.
    Then we have [G]≅[B_1]⊗⋯⊗[B_n].






§ PSEUDO-GORENSTEIN EDGE RINGS

In this section, we study the pseudo-Gorensteinness of edge rings.



 §.§ Preliminary on graph theory

Before that, we need some notions and notations on (directed) graphs.
Let G be a 2-connected graph.
Then G has an (open) ear decomposition (cf. <cit.>), i.e., G can be decomposed as C∪ P_1∪⋯∪ P_r where C is a cycle, P_i is a path and (C∪ P_1 ∪⋯∪ P_i-1)∩ P_i consists of end vertices of P_i for each i.
Let ϕ(G) denote the minimum number of even paths (containing the first cycle) in an ear decomposition of G.
Note that ϕ(G)≥ 1 if G is bipartite since the first cycle must be an even
cycle.
We call an ear decomposition of G optimal if the number of even paths is just ϕ(G).
Let G be a 2-connected bipartite graph and fix an optimal ear decomposition G=C∪ P_1∪⋯∪ P_r with

    V(P_i)=p_i,0,p_i,1,…,p_i,m_i,    V(C)=c_1,…,c_2n,                      
        E(P_i)=p_i,0,p_i,1,p_i,1,p_i,2,…,p_i,m_i-1,p_i,m_i and
        E(C)=c_1,c_2,c_2,c_3,…,c_2n-1,c_2n,c_2n,c_1.

We set

    E_i {{p_i,1,p_i,2},{p_i,3,p_i,4},⋯,{p_i,m_i -2,p_i,m_i-1}}    if m_i is odd,  
    {{p_i,1,p_i,2},{p_i,3,p_i,4},⋯,{p_i,m_i-1,p_i,m_i}}    if m_i is even,
            
        E_C {{c_1,c_2},{c_3,c_4},⋯,{c_2n-1,c_2n}} and
        E_C∪⋃_i=1^rE_i.

Note that E_i=∅ if m_i=1, and that =(ϕ(G)+V(G)-1)/2.



The following terminologies and facts are mentioned in <cit.>:
For a directed graph D,

    
  * the set of edges entering a subset W⊂ V(D) is called a directed cut if no edges leave W.
    
  * A directed graph D is strongly connected if there is a directed path from every vertex to any other.
    
  * The following are equivalent:
          
              
  * D is strongly connected;
              
  * D does not contain directed cuts;
              
  * D has a so-called directed ear decomposition, which is an ear decomposition of D, denoted as <Ref>, such that each edge c_i,c_i+1 (resp. p_i,j,p_i,j+1) is directed from c_i to c_i+1 (resp. p_i,j to p_i,j+1).
          


For a 2-connected bipartite graph G with the partition V(G)=V_1⊔ V_2, let G' be the directed graph obtained from G by orienting each edge from V_1 to V_2
and let G' be the directed graph obtained by reversing the orientation of edges  of G'.
We can see that G' has a directed ear decomposition.
Let σ(G) denote the minimum cardinality of a subset of E(G) that contains at least one edge of each directed cut of G'.

According to <cit.> and <cit.>, we have the following:


    Let G be a 2-connected bipartite graph.
    Then we have
    
    ℓ_G=σ(G)==ϕ(G)+V(G)-1/2.



A graph G is called matching-covered if it is connected and each edge is contained in some perfect matching.
There are many characterizations of matching-covered bipartite graphs (see, e.g., <cit.> or <cit.>), of which the following is the most important one for this paper:


    Let G be a connected bipartite graph with the partition V(G)=V_1⊔ V_2.
    Then the following are equivalent:
    
        
  * G is matching-covered;
        
  * V_1=V_2 and N_G(T)>T for every non-empty subset T⊊ V_1;
        
  * G is 2-connected and ϕ(G)=1.
    



    If G is not bipartite, then (i) and (iii) in <Ref> are not equivalent, but the implication “(i) ⟹ (iii)” is true in general (see <cit.>).


For a graph G, we say that G is k-regular if all vertices of G have the same degree k, and G is regular if G is k-regular for some k.


    
    Let G be a connected k-regular graph.
    Then for an independent set T⊂ V(G),
    one has T = N_G(T) if and only if G = B(T).



    Since G is k-regular, we have
    
    k T = ∑_v ∈ N_G(T)_B(T)(v) ≤∑_v ∈ N_G(T) k = k N_G(T),

    where _B(T)(v) is the degree of v in the subgraph B(T).
    If T = N_G(T), then the equality of <Ref> holds, which implies that G=B(T) since G is connected.

    Conversely, assume that G = B(T).
    In this case, G is bipartite with the partition V(G) = T ⊔ N_G(T).
    Moreover, the equality of <Ref> holds, so we have
    T = N_G(T).






 §.§ The case of bipartite graphs

In this subsection, we investigate when the edge rings of bipartite graphs are pseudo-Gorenstein.
Let G be a bipartite graph with the partition V_1⊔ V_2 and let B_1,…,B_m be the blocks of G.
Then we have h_s([G])=∏_i=1^m h_s([B_i]) from <Ref>, so it is enough to study the case when G is 2-connected.
In what follows, we assume G is 2-connected, fix an optimal ear decomposition described as <Ref> and let  be the subset of E(G) defined in <Ref>.

We first give the following lemma:


    We have v_∈∫(S_G).


    From <Ref>, it is enough to show that v_∈ H_i^> for each i∈ [d] and v_∈ H_T^> for any acceptable set T⊊ V_1.
    Since ⋃_e∈e=V(G),
    we have v_∈ H_i^> for any i∈ V(G).
    For an acceptable set T⊊ V_1, let W  T∪ N_G(T) and v,w∈ E(G): v∈ V_1∖ T  and  w∈ N_G(T).
    Then  is a directed cut of G', but is not a directed cut of G' since G' has a directed ear decomposition, and hence G' does not contain directed cuts.
    This implies that ∩≠∅, and hence v_∈ H_T^> from <Ref>.


We now present the following main theorem in this subsection:


    Let G be a 2-connected bipartite graph.
    Then the following are equivalent:
    
        
  * [G] is pseudo-Gorenstein;
        
  * ϕ(G)=1;
        
  * G is matching-covered.
    



    It follows from <Ref> that (ii) and (iii) are equivalent, so it is enough to show that (i) and (ii) are equivalent.

    First, we show that (i) implies (ii).
    Suppose that ϕ(G)≥ 2.
    Then there exists i such that P_i has the even length.
    Let E'_i p_i,0,p_i,1,p_i,2,p_i,3,…,p_i,m_i-2,p_i,m_i-1
    and let '  (∖ E_i)∪ E'_i.
    We can see that v_≠ v_' and v_,v_'∈∫(S_G) from <Ref>.
    In particular, we have v_,v_'∈∫(ℓ_G P_G)∩^d
    since ='=ℓ_G from <Ref>.
    Therefore, we get h_s= (ω_[G])_ℓ_G=∫(ℓ_G P_G)∩^d≥ 2.

    Next, we prove that (ii) implies (i).
    Actually, this has been already shown in <cit.> via interior polynomials, but we will provide a short self-contained proof of it.
    In this case, we can easily see that v_=(1,…,1) and this is the unique lattice point in ∫(ℓ_G P_G).
    Thus, we have h_s=1.


<Ref> gives some corollaries.


    Let G be a bipartite graph and let B_1,…, B_m be the blocks of G.
    Then [G] is pseudo-Gorenstein if and only if B_i is matching-covered for each i=1,…,m.



    It follows immediately from h_s([G])=∏_i=1^m h_s([B_i]).


Let G be a graph.
A cycle containing all the vertices of G is said to be a Hamilton cycle, and a graph containing a Hamilton cycle is said to be Hamiltonian.


    Let G be a Hamiltonian bipartite graph.
    Then [G] is pseudo-Gorenstein.



    In this situation, we have ϕ(G)=1.
    Indeed, G has an optimal ear decomposition C∪ P_1∪⋯ P_r where C is a Hamilton cycle of G and P_i is a path of length 1.



    Let G be a connected regular bipartite graph with the partition V_1⊔ V_2.
    Then [G] is pseudo-Gorenstein.



    From <Ref>, it is enough to show that V_1=V_2 and N_G(T)>T for every non-empty subset T⊊ V_1.
    The fact that G is regular implies that G has a perfect matching (and hence V_1=V_2) and that N_G(T)≥T for every subset T⊂ V_1 (<cit.>).
    If N_G(T)=T holds, then we have T=V_1 from <Ref>, thus N_G(T)>T for every non-empty subset T⊊ V_1.



    Let G be a 2-connected bipartite graph with the partition V_1⊔ V_2.
    If [G] is pseudo-Gorenstein, then we have V_1=V_2 (in particular, V(G) is even).



    It follows immediately from <Ref>.


We summarize the results of <Ref> in the following figure:







 §.§ The case of non-bipartite graphs

In this subsection, we investigate the pseudo-Gorensteinness of the edge rings of non-bipartite graphs.
Actually, most of the implications in <Ref> are no longer true.
We first give their counterexamples in the case of 2-connected non-bipartite graphs satisfying the odd cycle condition.


    (i) Let H_1 be the graph on the vertex set V(H_1)=1,2,3,4 with the edge set E(H_1)=1,2,2,3,3,4,1,4,1,3 (see <Ref>).
    This graph is not matching-covered since there is no perfect matching containing 1,3.
    On the other hand, we can compute h([H_1];t)=1+t, so H_1 is a counterexample to the implication  “pseudo-Gorenstein ⟹ matching-covered”.

    

    (ii) Let H_2 be the graph on the vertex set V(H_2)=1,2,3,4,5,6,7 with the edge set E(H_2)=1,2,2,3,3,4,4,5,1,5,1,6,3,6,3,7,5,7 (see <Ref>).
    We can see that ϕ(H_2)=2, V(H_2) is odd and h([H_2];t)=1+2t+t^2.
    Thus, H_2 is a counterexample to the implications “pseudo-Gorenstein ⟹ ϕ(G)=1” and “pseudo-Gorenstein ⟹ even number of vertices”.
    

    (iii) Let H_3 be the graph on the vertex set V(H_3)=1,2,3,4,5,6 with the edge set E(H_3)=1,2,2,3,3,4,4,5,5,6,1,6,1,3,3,5,3,6 (see <Ref>).
    This graph is Hamiltonian, and
    we can see that ϕ(H_3)=1 and h([H_3];t)=1+3t+3t^2.
    Therefore, H_3 is a counterexample to the implications “ϕ(H_3)=1 ⟹ pseudo-Gorenstein” and “Hamiltonian ⟹ pseudo-Gorenstein”.

    

    (iv) Consider the complete graph K_5 with five vertices.
    This graph is regular, but we can compute h([K_5];t)=1 + 5t + 5t^2.
    This implies that K_5 is a counterexample to the implication “regular ⟹ pseudo-Gorenstein”.




In particular, each direction of the implication (3) in <Ref> is incorrect.
Moreover, the following example shows that the value of ϕ(G) does not depend on the pseudo-Gorensteinness of edge rings, that is, for any k∈_>0, there exists a 2-connected graph G satisfying the odd cycle condition with h_s([G])=1 and ϕ(G)=k:

    For k∈_>0, let _k be the graph on the vertex set V(_k) v_1,…,v_k+1∪u_1,…,u_k∪w_1,…,w_k with the edge set
    
    E(_k) v_1,v_k+1∪⋃_i=1^k v_i,u_i,v_i,w_i,u_i,v_i+1,w_i,v_i+1

    (see <Ref>).
    
    Then _k is a 2-connected graph satisfying the odd cycle condition with ϕ(_k)=k.
    Indeed, any odd cycle of _k contains the edge v_1,v_k+1, so _k satisfies the odd cycle condition.
    Moreover, we can easily see that for any ear decomposition of _k, the number of even paths appearing in it is just k, which implies that ϕ(_k)=k.

    We show that [_k] is pseudo-Gorenstein.
    From <cit.>, we can see that
    
    I__k=b_i  x_v_i,u_ix_w_i,v_i+1-x_u_i,v_i+1x_v_i,w_i : i ∈ [k].

    Since these binomials b_i have no common variables, we obtain
    
    [_k] ≅⊗_i=1^k [x_v_i,u_i,x_w_i,v_i+1,x_u_i,v_i+1,x_v_i,w_i]/(b_i)[x_v_1,v_k+1]

    and h([_k];t)=(1+t)^k, which implies that h_s([_k])=1.


The pseudo-Gorensteinness of the edge rings of non-bipartite graphs is quite different from that of bipartite graphs, and it seems difficult to completely characterize it.

We end this section by giving the following two sufficient conditions for edge rings to be pseudo-Gorenstein:

    Let G be a matching-covered connected graph satisfying the odd cycle condition.
    Then [G] is pseudo-Gorenstein.



    Our assertion holds from <Ref> if G is bipartite, thus we may assume that G is not bipartite.
    Let v=(1,…,1), then v belongs to S_G since G has a perfect matching.
    We show that v is in the unique lattice point in ∫(ℓ_G P_G).
    From <Ref>, it suffices to prove that v∈ H_T^> for any independent set T of G.

    Since G is a non-bipartite connected graph, there exists f∈ E(G)∖ E(B(T)) with f∩ N_G(T)≠∅.
    We can take a perfect matching  including f since G is matching-covered, and hence v_=v∈ H_T^> from <Ref>.



    Let G be a connected regular graph with an even number of vertices
    satisfying the odd cycle condition.
    Then [G] is pseudo-Gorenstein.



    Our assertion holds from <Ref> if G is bipartite, thus we may assume that G is not bipartite.
    As in the proof of <Ref>, it is enough to show that (1,…,1) ∈ H_T^> for any independent set T of G.
    We obtain T≤N_G(T) by <Ref>.
    If T = N_G(T) holds, then we have G = B(T) from <Ref>,
    so G is bipartite, which contradicts with the assumption.
    Therefore, for any independent set T, we have T < N_G(T), and the point (1, …, 1) belongs to H_T^>.






§ ALMOST GORENSTEIN EDGE RINGS




 §.§ A new family of almost Gorenstein edge rings

In this subsection, we consider the following graph:
For integers m, n ≥ 3 and 0 ≤ r ≤min{m, n}, let G_m, n, r be the graph on the vertex set V(G_m, n, r)=[m + n] with the edge set
E(G_m, n, r) = {i, j + m} : i ∈ [m], j ∈ [n]∖1, 1 + m, 2, 2 + m, …, r , r + m.
We can easily see that V(G_m,n,r) has the partition V(G_m,n,r)=V_1⊔ V_2, where V_1 1,…,m and V_2 m+1,…,m+n, and G_m,n,r is bipartite.
Moreover, G_m,n,r is 2-connected and the subset T⊂ V_1 is an acceptable set if and only if T=i for some i∈ [r].

To compute the h-polynomial of [G_m,n,r], we recall the h-polynomial of the edge ring of a complete bipartite graph.

    Let K_m,n denote the complete bipartite graph with m+n vertices. Then
    
    h([K_m,n];t)=∑_i=0^min{m,n}m-1in-1it^i.




    We have
    
    h([G_m, n, r]; t) = 1 + (m - 1)(n - 1) - r t
            + ∑_i = 2^min{m, n}m - 1in - 1i t^i.




    By <Ref>, it suffices to show that
    
    L_P_K_m, n(k) - L_P_G_m, n, r(k)
            = k P_K_m, n∩^m + n - k P_G_m, n, r∩^m + n
            = r k + m + n - 3m + n - 2.

    For each i = 1, …, r, let
    
    A_i (x_1, …, x_m + n) ∈^m + n_≥ 0x_i + x_m + i≥ k + 1, ∑_j = 1^m x_j = ∑_j = 1^n x_m + j = k

    and A  A_1∪⋯∪ A_r.
    Note that A_i ∩ A_j = ∅ if i  j.
    We show that A ⊂ k P_K_m, n∩^m + n and
    
    A_i=k + m + n -3m + n - 2 for each i, and hence A = r k + m + n -3m + n - 2.

    Let S_i  [m+n] ∖i for each i ∈ [r] and let σ_i  S_i →^m+n be the map defined as follows:
    
    σ_i(l) _l+_m+i    if 1 ≤ l ≤ m,        
    _i+_l         if m + 1 ≤ l ≤ m + n for l∈ S_i.

    Moreover, we denote the set of all (k-1)-element multisubsets on S_i by U_i and define the map f_i  U_i→^m + n as
    
    f_i(u) _i + _m+i + ∑_l∈ uσ(l)   for u ∈ U_i.

    Then we can see that f_i(U_i) = A_i and f_i is a bijection on A_i.
    Therefore, we have A_i ⊂ k P_K_m, n∩^m + n and
    A_i = U_i = k+m+n-3m+n-2.

    It remains to show that
    (k P_K_m, n∩^m+n) ∖ A = k P_G_m, n, r∩^m+n.
    Any a = (a_1, …, a_m + n) ∈ A_i does not satisfy the inequality ∑_j  i a_m + j - a_i ≥ 0 for any i∈ [r].
    Hence a ∉ H_i^+, and consequently a ∉ kP_G_m, n, r∩^m+n.
    Conversely, for any a = (a_1, …, a_m+n) ∈ (kP_K_m, n∩^m+n) ∖ A,
    the element a can be expressed as
    
    a = ∑_i ∈ [m], j ∈ [n] b_i, j (_i + _m + j)  for some b_i, j∈_≥ 0.

    Since (_i + _m + i) + (_j + _m + j) = (_i + _m + j) + (_j + _m + i),
    we may assume b_i, i≥ 0 and b_j, j = 0 for j ∈ [m] ∖{i}.
    Suppose b_i, i > 0. Then there exist j, l ∈ [m] ∖{i} such that b_j, l > 0 since a_i + a_m + i≤ k,
    and we have (_i + _m + i) + (_j + _m + l) = (_i + _m + l) + (_j + _m + i).
    Hence we may assume b_i, m+i = 0, and therefore we have a ∈ kP_G_m, n, r∩^m+n.



    We have μ(ω_[G_n,n,r])≥ r(n-3)+1.



    It is enough to show that the set of the lattice points in ∫(S_G_n,n,r) corresponding to the minimal generators of ω_[G_n,n,r] contains the following:
    
    v_i,j (1,…,1,iǰ,1,…,1,i+nǰ,1,…,1)  for   i∈{1,…, n} and  j∈{1,…,n-2}.

    We can see that v_i,j belongs to ∫(S_G_n,n,r) since v_i,j∈ H_k^> for any k∈ [2n] and v_i,j∈ H_l^> for any l∈ [r].
    If we can write v_i,j=v'+ρ(e) for some v'∈∫(S_G_n,n,r) and e∈ E(G_n,n,r), then e=i,i+n since v'∈ H_k^> for any k∈ [2n], a contradiction to i,i+n∉ E(G_n,n,r).
    Therefore, v_i,j cannot be written as a sum of an element in
    ∫(S_G_n,n,r) and an element in S_G_n,n,r∖0, which is the desired result.



    The edge ring [G_m,n,r] is almost Gorenstein if and only if m=n.



    If [G_m,n,r] is almost Gorenstein, then we have h_s([G_m,n,r])=1 from <Ref>, so we get m=n.

    Suppose that m=n.
    It follows from <Ref> and <Ref> that
    
    r(n-3)=e([G_n,n,r]) ≥μ(ω_R)-1 ≥ r(n-3).

    Therefore, we have e([G_n,n,r])=μ(ω_R)-1 and [G_n,n,r] is almost Gorenstein from <Ref>.




 §.§ Observations and questions on almost Gorenstein edge rings


<Ref> and <Ref> tell us that [G_n,n,r] is almost Gorenstein and its h-vector (h_0,h_1,…,h_s) satisfies the following condition:

    h_i=h_s-i for  i=0,2,3,…,s/2. *

As far as the authors know, the h-vectors of almost Gorenstein edge rings discovered so far satisfy the condition <Ref>.
For example, the edge ring of a complete graph K_2m is almost Gorenstein (<cit.>), its h-vector has been computed (<cit.>) and satisfies condition <Ref>.

Moreover, the edge ring of a complete multipartite graph K_1,n,n (n≥ 2) is almost Gorenstein (<cit.>), which is isomorphic to a certain Hibi ring (<cit.>).
The h-vector of this Hibi ring satisfies condition <Ref> (<cit.>).

Furthermore, the h-vectors of the edge rings of a certain family of graphs _n, consisting of n triangles that share a single common vertex, have been investigated in <cit.>.
According to <cit.>, [_n] is almost Gorenstein and its h-vector satisfies condition <Ref>.

These results naturally pose us with the following question:

    Do the h-vectors of almost Gorenstein edge rings satisfy condition <Ref>?

Actually, the following example gives a negative answer to this question:


    Let  be the Petersen graph (see <Ref>).
    
    We can see that  satisfies the odd cycle condition.
    Moreover, we can check
    
    h([];t)=1+5t+15t^2+25t^3+5t^4+t^5

    by using  (<cit.>).
    It follows from <cit.> that [] is not Gorenstein but almost Gorenstein.
    Moreover, this h-vector does not satisfy condition <Ref>.


We can still see that the h-vectors of almost
Gorenstein edge rings are “almost symmetric”, which means that for i=0,…,s/2, the equality h_i=h_s-i holds for all but at most one value of i.
Unfortunately, that is also not true in general.


    Let W_10 be the wheel graph with 10 vertices (see <Ref>).
    
    This graph satisfies the odd cycle condition and
     tells us that the h-polynomial of [W_10] is
    
    h([W_10];t)=1+8t+27t^2+30t^3+9t^4+t^5

    and that μ(ω_[W_10])=7.
    Therefore, we have e([W_10])=μ(ω_[W_10])-1=6, and hence [W_10] is almost Gorenstein from <Ref>.


While these counterexamples exist, we have yet to find the edge ring of a “bipartite graph” that is almost Gorenstein and whose h-vector does not satisfy condition <Ref>.


    Do the h-vectors of the almost Gorenstein edge rings of bipartite graphs satisfy condition <Ref>?


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