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We give a lower bound for the Castelnuovo-Mumford regularity of binomial edge ideals of block graphs by computing the two distinguished extremal Betti numbers of a new family of block graphs, called flower graphs. Moreover, we present linear time algorithms to compute the Castelnuovo-Mumford regularity and the Krull dimension of binomial edge ideals of block graphs.

The Hilbert function, its generating function and the Hilbert polynomial of a graded ring K[x 1 , . . . , x k ] have been extensively studied since the famous paper of Hilbert: Ueber die Theorie der algebraischen Formen [Hil90]. In particular, the coefficients and the degree of the Hilbert polynomial play an important role in Algebraic Geometry. If the ring grading is non-standard, then its Hilbert function is not eventually equal to a polynomial but to a quasi-polynomial. It turns out that a Hilbert quasi-polynomial P of degree n splits into a polynomial S of degree n and a lower degree quasi-polynomial T . We have completely determined the degree of T and the first few coefficients of P . Moreover, the quasi-polynomial T has a periodic structure that we have described. We have also implemented a software to compute effectively the Hilbert quasi-polynomial for any ring K[x 1 , . . . , x k ]/I. Keywords:Non-standard gradings, Hilbert quasi-polynomial. and the Hilbert-Poincaré series of (R/I, W ) is given byWhen the grading given by W is clear from the contest, we denote respectively the Hilbert function and the Hilbert-Poincaré series of (R/I, W ) by H R/I and HP R/I . The following two well-known results characterize the Hilbert-Poincaré series and the Hilbert function, but the latter holds only for standard grading.Theorem 1 (Hilbert-Serre) The Hilbert-Poincaré series of (R/I, W ) is a rational function, that is for suitable h(t) ∈ Z[t] we have thatThe polynomial h(t) which appears in the denominator of HP R/I is called hvector and we denote it by < I >.Definition 2 Let R be a polynomial standard graded ring and I a homogeneous ideal of R. Then there exists a polynomial P R/I (x) ∈ Q[x] such thatThis polynomial is called Hilbert polynomial of R/I. Hilbert quasi-polynomialsAs we will see, in the case of non-standard grading, the Hilbert function of (R/I, W ) is definitely equal to a quasi-polynomial P W R/I instead of a polynomial. The main aim of this section is to investigate the structure of P W R/I , such as its degree, its leading coefficient and proprieties of its coefficients. Existence of the Hilbert quasi-polynomialsWe recall that a function f : N → N is a (rational) quasi-polynomial of period s if there exists a set of s polynomials {p 0 , . . . , p s−1 } in Q[x] such that we have f (n) = p i (n) when n ≡ i mod s. Let d := lcm(d 1 , . . . , d k ). We are going to show that the Hilbert function of (R/I, W ) is definitely equal to a quasi-polynomial of period d.Proposition 3 Let (R/I, W ) be as above. There exists a unique quasi-polynomial P W R/I := {P 0 , . . . , P d−1 } of period d such that H R/I (n) = P W R/I (n) for all n ≫ 0, that is H R/I (n) = P i (n) ∀i ≡ n mod d and ∀n ≫ 0 P W R/I is called the Hilbert quasi-polynomial associated to (R/I,W).Proof. From the Hilbert-Serre theorem, we know that the Hilbert-Poincaré series of R/I can be written as a rational function

We describe the simplicial complex ∆ such that the initial ideal of J G is the Stanley-Reisner ideal of ∆. By ∆ we show that if J G is (S 2 ) then G is accessible. We also characterize all accessible blocks with whiskers of cycle rank 3 and we define a new infinite class of accessible blocks with whiskers for any cycle rank. Finally, by using a computational approach, we show that the graphs with at most 12 vertices whose binomial edge ideal is Cohen-Macaulay are all and only the accessible ones. https://mathworld.wolfram.com/HelmGraph.html.

In this work, we provide a necessary and sufficient condition on a polyomino ideal for having the set of inner 2-minors as graded reverse lexicographic Gröbner basis, due to combinatorial properties of the polyomino itself. Moreover, we prove that when the latter holds the ideal coincides with the lattice ideal associated to the polyomino, that is the ideal is prime. As an application, we describe two new infinite families of prime polyominoes.