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Register a new userBinomial edge ideals and conditional independence statements
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We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Grobner basis in a lexicographic order induced by a vertex labeling. Such graphs are chordal and claw-free. We give a reduced squarefree Grobner basis for general $G$. It follows that all binomial edge ideals are radical ideals. Their minimal primes can be characterized by particular subsets of the vertices of $G$. We provide sufficient conditions for Cohen--Macaulayness for closed and nonclosed graphs. Binomial edge ideals arise naturally in the study of conditional independence ideals. Our results apply for the class of conditional independence ideals where a fixed binary variable is independent of a collection of other variables, given the remaining ones. In this case the primary decomposition has a natural statistical interpretation
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This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Grobner basis can be computed by studying paths in the graph. Since these Grobner bases are square-free, generalized binomial edge ideals are radical. To find the primary decomposition a combinatorial problem involving the connected components of subgraphs has to be solved. The irreducible components of the solution variety are all rational.
We determine the Castelnuovo-Mumford regularity of binomial edge ideals of complement reducible graphs (cographs). For cographs with $n$ vertices the maximum regularity grows as $2n/3$. We also bound the regularity by graph theoretic invariants and construct a family of counterexamples to a conjecture of Hibi and Matsuda.
We study a class of determinantal ideals that are related to conditional independence (CI) statements with hidden variables. Such CI statements correspond to determinantal conditions on a matrix whose entries are probabilities of events involving the observed random variables. We focus on an example that generalizes the CI ideals of the intersection axiom. In this example, the minimal primes are again determinantal ideals, which is not true in general.
Let $G$ be a connected simple graph on the vertex set $[n]$. Banerjee-Betancourt proved that $depth(S/J_G)leq n+1$. In this article, we prove that if $G$ is a unicyclic graph, then the depth of $S/J_G$ is bounded below by $n$. Also, we characterize $G$ with $depth(S/J_G)=n$ and $depth(S/J_G)=n+1$. We then compute one of the distinguished extremal Betti numbers of $S/J_G$. If $G$ is obtained by attaching whiskers at some vertices of the cycle of length $k$, then we show that $k-1leq reg(S/J_G)leq k+1$. Furthermore, we characterize $G$ with $reg(S/J_G)=k-1$, $reg(S/J_G)=k$ and $reg(S/J_G)=k+1$. In each of these cases, we classify the uniqueness of extremal Betti number of these graphs.
Let $G$ be a finite simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in the polynomial ring $S = K[x_1, ldots, x_n, y_1, ldots, y_n].$ In this article, we compute the Hilbert series of binomial edge ideal of decomposable graphs in terms of Hilbert series of its indecomposable subgraphs. Also, we compute the Hilbert series of binomial edge ideal of join of two graphs and as a consequence we obtain the Hilbert series of complete $k$-partite graph, fan graph, multi-fan graph and wheel graph.
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