Let I(G) be the edge ideal associated to a simple graph G. We study the graded Betti numbers that appear in the linear strand of the minimal free resolution of I(G).

Lower bounds are given for the depths of R/I for t ≥ 1 when I is the edge ideal of a tree or forest. The bounds are given in terms of the diameter of the tree, or in case of a forest, the largest diameter of a connected component and the number of connected components. These lower bounds provide a lower bound on the power for which the depths stabilize.

Some new relations in the family of edge ideal metrics for biopolymers are proved. Also, some statistical relations among the metrics are shown using a big synthetic contact structure set; for unary contact structures, the relations are linear with a high significance.

We determine the arithmetical rank of every edge ideal of a Ferrers graph.

We discuss the linearity of the minimal free resolution of a power of an edge ideal.

An analytic expression for the limiting current in the vicinity of metallic and dielectric edges is developed. It is shown that the basic V law remains valid and, for large dielectric coefficients (er.5), the average limiting current density is not explicitly dependent on the angle of the metallic edge. Moreover, the limiting current depends primarily on the typical curvature of the emitter and...

Let G be a finite graph allowing loops, having no multiple edge and no isolated vertex. We associate G with the edge polytope PG and the toric ideal IG. By classifying graphs whose edge polytope is simple, it is proved that the toric ideals IG of G possesses a quadratic Gröbner basis if the edge polytope PG of G is simple. It is also shown that, for a finite graph G, the edge polytope is simple...