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Let $G$ be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate $G$ with the edge polytope ${cal P}_G$ and the toric ideal $I_G$. By classifying graphs whose edge polytope is simple, it is proved that the toric ideals $I_G$ of $G$ possesses a quadratic Grobner basis if the edge polytope ${cal P}_G$ of $G$ is simple. It is also shown that, for a finite graph $G$, the edge polytope is simple but not a simplex if and only if it is smooth but not a simplex. Moreover, the Ehrhart polynomial and the normalized volume of simple edge polytopes are computed.
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In the present paper, we consider the problem when the toric ring arising from an integral cyclic polytope is Cohen-Macaulay by discussing Serres condition and we give a complete characterization when that is Gorenstein. Moreover, we study the normality of the other semigroup ring arising from an integral cyclic polytope but generated only with its vertices.
It is known that the coordinate ring of the Grassmannian has a cluster structure, which is induced from the combinatorial structure of a plabic graph. A plabic graph is a certain bipartite graph described on the disk, and there is a family of plabic graphs giving a cluster structure of the same Grassmannian. Such plabic graphs are related by the operation called square move which can be considered as the mutation in cluster theory. By using a plabic graph, we also obtain the Newton-Okounkov polytope which gives a toric degeneration of the Grassmannian. The purposes of this article is to survey these phenomena and observe the behavior of Newton-Okounkov polytopes under the operation called the combinatorial mutation of polytopes. In particular, we reinterpret some operations defined for Newton-Okounkov polytopes using the combinatorial mutation.
In this paper we study monomial ideals attached to posets, introduce generalized Hibi rings and investigate their algebraic and homological properties. The main tools to study these objects are Groebner basis theory, the concept of sortability due to Sturmfels and the theory of weakly polymatroidal ideals.
The Jacobian algebra $mathsf{A}$ arising from a consistent dimer model is derived equivalent to crepant resolutions of a $3$-dimensional Gorenstein toric singularity $R$, and it is also called a non-commutative crepant resolution of $R$. This algebra $mathsf{A}$ is a maximal Cohen-Macaulay (= MCM) module over $R$, and it is a finite direct sum of rank one MCM $R$-modules. In this paper, we observe a relationship between properties of a dimer model and those of MCM modules appearing in the decomposition of $mathsf{A}$ as an $R$-module. More precisely, we take notice of isoradial dimer models and divisorial ideals which are called conic. Especially, we investigate them for the case of $3$-dimensional Gorenstein toric singularities associated with reflexive polygons.
Let $G$ be a finite simple graph on the vertex set $V(G) = {x_1, ldots, x_n}$ and $I(G) subset K[V(G)]$ its edge ideal, where $K[V(G)]$ is the polynomial ring in $x_1, ldots, x_n$ over a field $K$ with each ${rm deg} x_i = 1$ and where $I(G)$ is generated by those squarefree quadratic monomials $x_ix_j$ for which ${x_i, x_j}$ is an edge of $G$. In the present paper, given integers $1 leq a leq r$ and $s geq 1$, the existence of a finite connected simple graph $G = G(a, r, d)$ with ${rm im}(G) = a$, ${rm reg}(R/I(G)) = r$ and ${rm deg} h_{K[V(G)]/I(G)} (lambda) = s$, where ${rm im}(G)$ is the induced matching number of $G$ and where $h_{K[V(G)]/I(G)} (lambda)$ is the $h$-polynomial of $K[V(G)]/I(G)$.
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