Ehrhart clutters: Regularity and Max-Flow Min-Cut

If C is a clutter with n vertices and q edges whose clutter matrix has column vectors V={v1,...,vq}, we call C an Ehrhart clutter if {(v1,1),...,(vq,1)} is a Hilbert basis. Letting A(P) be the Ehrhart ring of P=conv(V), we are able to show that if A is the clutter matrix of a uniform, unmixed MFMC clutter C, then C is an Ehrhart clutter and in this case we provide sharp bounds on the Castelnuovo-Mumford regularity of A(P). Motivated by the Conforti-Cornuejols conjecture on packing problems, we conjecture that if C is both ideal and the clique clutter of a perfect graph, then C has the MFMC property. We prove this conjecture for Meyniel graphs, by showing that the clique clutters of Meyniel graphs are Ehrhart clutters. In much the same spirit, we provide a simple proof of our conjecture when C is a uniform clique clutter of a perfect graph. We close with a generalization of Ehrhart clutters as it relates to total dual integrality.Comment: Electronic Journal of Combinatorics, to appea

Normally torsion-free edge ideals of weighted oriented graphs

Let $I=I(D)$ be the edge ideal of a weighted oriented graph $D$, let $G$ be the underlying graph of $D$, and let $I^{(n)}$ be the $n$-th symbolic power of $I$ defined using the minimal primes of $I$. We prove that $I^2=I^{(2)}$ if and only if (i) every vertex of $D$ with weight greater than $1$ is a sink and (ii) $G$ has no triangles. As a consequence, using a result of Mandal and Pradhan, and the classification of normally torsion-free edge ideals of graphs, it follows that $I^n=I^{(n)}$ for all $n\geq 1$ if and only if (a) every vertex of $D$ with weight greater than $1$ is a sink and (b) $G$ is bipartite. If $I$ has no embedded primes, conditions (a) and (b) classify when $I$ is normally torsion-free. Using polyhedral geometry and integral closure, we give necessary conditions for the equality of ordinary and symbolic powers of monomial ideals with a minimal irreducible decomposition. Then, we classify when the Alexander dual of the edge ideal of a weighted oriented graph is normally torsion-free.Comment: We added some result

Analytical validation of an automated assay for the measurement of adenosine deaminase (ADA) and its isoenzymes in saliva and a pilot evaluation of their changes in patients with SARS-CoV-2 infection

Objectives The aim of the present study was to validate a commercially available automated assay for the measurement of total adenosine deaminase (tADA) and its isoenzymes (ADA1 and ADA2) in saliva in a fast and accurate way, and evaluate the possible changes of these analytes in individuals with SARS-CoV-2 infection. Methods The validation, in addition to the evaluation of precision and accuracy, included the analysis of the effects of the main procedures that are currently being used for SARS-CoV-2 inactivation in saliva and a pilot study to evaluate the possible changes in salivary tADA and isoenzymes in individuals infected with SARS-CoV-2. Results The automated assay proved to be accurate and precise, with intra- and inter-assay coefficients of variation below 8.2%, linearity under dilution linear regression with R2 close to 1, and recovery percentage between 80 and 120% in all cases. This assay was affected when the sample is treated with heat or SDS for virus inactivation but tolerated Triton X-100 and NP-40. Individuals with SARS-CoV-2 infection (n=71) and who recovered from infection (n=11) had higher mean values of activity of tADA and its isoenzymes than healthy individuals (n=35). Conclusions tADA and its isoenzymes ADA1 and ADA2 can be measured accurately and precisely in saliva samples in a rapid, economical, and reproducible way and can be analyzed after chemical inactivation with Triton X-100 and NP-40. Besides, the changes observed in tADA and isoenzymes in individuals with COVID-19 open the possibility of their potential use as non-invasive biomarkers in this disease