arXiv subjects
Publications and source records attributed to Giancarlo Rinaldo.
We provide the regularity and the Cohen-Macaulay type of binomial edge ideals of Cohen-Macaulay cones, and we show the extremal Betti numbers of some classes of Cohen-Macaulay binomial edge ideals: Cohen-Macaulay bipartite and fan graphs. In addition, we compute the Hilbert-Poincaré series of the binomial edge ideals of some Cohen-Macaulay bipartite graphs.
Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$ and let $Δ$ be its independence complex. We describe the well-covered circulant graphs with 2-dimensional $Δ$ and construct an infinite family of vertex-decomposable circulant graphs within this family.
Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$. We study the reduced Euler characteristic $\tildeχ$ of the independence complex $Δ(G)$ for $n=p^k$ with $p$ prime and for $n=2p^k$ with $p$ odd prime, proving that in both cases $\tildeχ$ does not vanish. We also give an example of circulant graph whose independence complex has $\tildeχ$ equals to $0$, giving a negative answer to R. Hoshino.
We compute one of the distinguished extremal Betti number of the binomial edge ideal of a block graph, and classify all block graphs admitting precisely one extremal Betti number.
We present an application of Hilbert quasi-polynomials to order domains, allowing the effective check of the second order-domain condition in a direct way. We also provide an improved algorithm for the computation of the related Hilbert quasi-polynomials. This allows to identify order domain codes more easily.
We classify the Cohen-Macaulay binomial edge ideals of cactus and bicyclic graphs.
Let $G$ be the circulant graph $C_n(S)$ with $S\subseteq\{ 1,\ldots,\left \lfloor\frac{n}{2}\right \rfloor\}$ and let $I(G)$ be its edge ideal in the ring $K[x_0,\ldots,x_{n-1}]$. Under the hypothesis that $n$ is prime we : 1) compute the regularity index of $R/I(G)$; 2) compute the Castelnuovo-Mumford regularity when $R/I(G)$ is Cohen-Macaulay; 3) prove that the circulant graphs with $S=\{1,\ldots,s\}$ are sequentially $S_2$ . We end characterizing the Cohen-Macaulay circulant graphs of Krull dimension $2$ and computing their Cohen-Macaulay type and Castelnuovo-Mumford regularity.
The BIX protocol is a blockchain-based protocol that allows distribution of certificates linking a subject with his public key, hence providing a service similar to that of a PKI but without the need of a CA. In this paper we analyze the security of the BIX protocol in a formal way, in four steps. First, we identify formal security assumptions which are well-suited to this protocol. Second, we present some attack scenarios against the BIX protocol. Third, we provide a formal security proof that some of these attacks are not feasible under our previously established assumptions. Finally, we show how another attack may be carried on.
In this note we prove that every closed graph $G$ is up to isomorphism a proper interval graph. As a consequence we obtain that there exist linear-time algorithms for closed graph recognition.
We classify all binomial edge ideals that are complete intersection and Cohen-Macaulay almost complete intersection. We also describe an algorithm and provide an implementation to compute primary decomposition of binomial edge ideals.
We discuss algebraic and homological properties of binomial edge ideals associated to graphs which are obtained by gluing of subgraphs and the formation of cones.
We classify the ideals of mixed products that are sequentially Cohen-Macaulay.
Let $S = K[x_1,..., x_n]$ be a polynomial ring over a field $K$. Let $I(G) \subseteq S$ denote the edge ideal of a graph $G$. We show that the $\ell$th symbolic power $I(G)^{(\ell)}$ is a Cohen-Macaulay ideal (i.e., $S/I(G)^{(\ell)}$ is Cohen-Macaulay) for some integer $\ell \ge 3$ if and only if $G$ is a disjoint union of finitely many complete graphs. When this is the case, all the symbolic powers $I(G)^{(\ell)}$ are Cohen-Macaulay ideals. Similarly, we characterize graphs $G$ for which $S/I(G)^{(\ell)}$ has (FLC). As an application, we show that an edge ideal $I(G)$ is complete intersection provided that $S/I(G)^{\ell}$ is Cohen-Macaulay for some integer $\ell \ge 3$. This strengthens the main theorem in [Effective Cowsik-Nori theorem for edge ideals by M.Crupi, G.Rinaldo, N.Terai, and K.Yoshida, Comm. Alg. 38 (2010), 3347-3357].
In this paper, we study several properties of the second power $I_Δ^2$ of a Stanley-Reisner ideal $I_Δ$ of any dimension. As the main result, we prove that $S/I_Δ$ is Gorenstein whenever $S/I_Δ^2$ is Cohen-Macaulay over any field $K$. Moreover, we give a criterion for the second symbolic power of $I_Δ$ to satisfy $(S_2)$ and to coincide with the ordinary power, respectively. Finally, we provide new examples of Stanley-Reisner ideals whose second powers are Cohen-Macaulay.
It is shown that any set of nonzero monomial prime ideals can be realized as the stable set of associated prime ideals of a monomial ideal. Moreover, an algorithm is given to compute the stable set of associated prime ideals of a monomial ideal.
Let $G$ be a simple graph on the vertex set $V(G) = [n] = \{1,...,n\}$ and edge ideal $E(G)$. We consider the class of closed graphs. A closed graph is a simple graph satisfying the following property: for all edges $\{i, j\}$ and $\{k, \ell\}$ with $i < j$ and $k < \ell$ one has $\{j, \ell\}\in E(G)$ if $i = k$, and $\{i, k\}\in E(G)$ if $j = \ell$. We state some criteria for the closedness of a graph $G$ that do not depend necessarily from the labelling of its vertex set. Consequently, if $S = K[x_1,..., x_n, y_1,..., y_n]$ is a polynomial ring in $2n$ variables with coefficients in a field $K$, we obtain some criteria for the Koszulness of the quotient algebra $S /J_G$, where $J_G$ is the binomial edge ideal of $S$ i.e. the ideal generated by the binomials $f_{ij} = x_iy_j - x_jy_i$ such that $i<j$ and $\{i,j\}$ is an edge of $G$ (\cite{HH}).
Let $I$ be a squarefree monomial ideal of a polynomial ring $S$. In this paper, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $S/I$ when one of the following conditions is satisfied: (1) $μ(I) \leq 5$; (2) $\arithdeg I \leq 4$.
A graph $G$ is well-covered if it has no isolated vertices and all the maximal independent sets have the same cardinality. If furthermore two times this cardinality is equal to $|V(G)|$, the graph $G$ is called very well-covered. The class of very well-covered graphs contains bipartite well-covered graphs. Recently in \cite{CRT} it is shown that a very well-covered graph $G$ is Cohen-Macaulay if and only if it is pure shellable. In this article we improve this result by showing that $G$ is Cohen-Macaulay if and only if it is pure vertex decomposable. In addition, if $I(G)$ denotes the edge ideal of $G$, we show that the Castelnuovo-Mumford regularity of $R/I(G)$ is equal to the maximum number of pairwise 3-disjoint edges of $G$. This improves Kummini's result on unmixed bipartite graphs.