Let I = (x v1 , . . . , x vq ) be a square-free monomial ideal of a polynomial ring K[x 1 , . . . , x n ] over an arbitrary field K and let A be the incidence matrix with column vectors v 1 , . . . , v q . We will establish some connections between algebraic properties of certain graded algebras associated to I and combinatorial optimization properties of certain polyhedra and clutters associated to A and I respectively. Some applications to Rees algebras and combinatorial optimization are presented. We study a conjecture of Conforti and Cornuéjols using an algebraic approach.
We study the family of graphs whose number of primitive cycles equals its cycle rank. It is shown that this family is precisely the family of ring graphs. Then we study the complete intersection property of toric ideals of bipartite graphs and oriented graphs. An interesting application is that complete intersection toric ideals of bipartite graphs correspond to ring graphs and that these ideals are minimally generated by Gröbner bases. We prove that any graph can be oriented such that its toric ideal is a complete intersection with a universal Gröbner basis determined by the cycles. It turns out that bipartite ring graphs are exactly the bipartite graphs that have complete intersection toric ideals for any orientation. 0 2000 Mathematics Subject Classification. Primary 05C75; Secondary 05C85, 05C20, 13H10.
Angiogenesis is an important adaptation mechanism of the blood vessels to the changing requirements of the body during development, aging, and wound healing. Angiogenesis allows existing blood vessels to form new connections or to reabsorb existing ones. Blood vessels are composed of a layer of endothelial cells (ECs) covered by one or more layers of mural cells (smooth muscle cells or pericytes). We constructed a computational Boolean model of the molecular regulatory network involved in the control of angiogenesis. Our model includes the ANG/TIE, HIF, AMPK/mTOR, VEGF, IGF, FGF, PLCγ/Calcium, PI3K/AKT, NO, NOTCH, and WNT signaling pathways, as well as the mechanosensory components of the cytoskeleton. The dynamical behavior of our model recovers the patterns of molecular activation observed in Phalanx, Tip, and Stalk ECs. Furthermore, our model is able to describe the modulation of EC behavior due to extracellular micro-environments, as well as the effect due to loss- and gain-of-function mutations. These properties make our model a suitable platform for the understanding of the molecular mechanisms underlying some pathologies. For example, it is possible to follow the changes in the activation patterns caused by mutations that promote Tip EC behavior and inhibit Phalanx EC behavior, that lead to the conditions associated with retinal vascular disorders and tumor vascularization. Moreover, the model describes how mutations that promote Phalanx EC behavior are associated with the development of arteriovenous and venous malformations. These results suggest that the network model that we propose has the potential to be used in the study of how the modulation of the EC extracellular micro-environment may improve the outcome of vascular disease treatments.
Let G be a simple graph with V G = n and no isolated vertices. Let be its stability number. We study invariants of the edge-ring of G that can be interpreted as invariants of G . If G has a cover by maximum stable sets we are able to prove the inequality ≤ n 2 . As a byproduct we prove that if G is vertex-critical, then ≤ n − A /2, where A is the intersection of all the minimum vertex covers of G . We estimate the smallest number of vertices in any maximal stable set of G to obtain a bound for the depth of the edge-ring of G .
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.