The aim of this work is to use linear programming and polyhedral geometry prove a duality formula for the ic-resurgence edge ideals. We show that ideal I clutter C I∨ blocker C∨ coincide. If bases certain uniform matroids, we recover resurgence I, if connected non-bipartite graph with perfect matching, Waldschmidt constant I∨.

A graph is closed when its vertices have a labeling by [n] with a certain property first discovered in the study of binomial edge ideals. In this article, we explore various aspects of closed graphs, including the number of closed labelings and clustering coefficients.

Let G be a finite graph and K[G] the edge ring of G. Based on the technique of Gröbner bases and initial ideals, it will be proved that, given integers f and d with 7 ≤ f ≤ d, there exists a finite graphG on [d] = {1, . . . , d} with depthK[G] = f and with Krull-dimK[G] = d.

Let R be a commutative ring and Max (R) be the set of maximal ideals of R. The regular digraph of ideals of R, denoted by −−→ Γreg(R), is a digraph whose vertex set is the set of all non-trivial ideals of R and for every two distinct vertices I and J , there is an arc from I to J whenever I contains a J-regular element. The undirected regular (simple) graph of ideals of R, denoted by Γreg(R), h...

Let $D$ be a weighted oriented graph and $I(D)$ its edge ideal. If contains an induced odd cycle of length $2n+1$, under certain condition we show that $ {I(D)}^{(n+1)} \neq {I(D)}^{n+1}$. We give necessary sufficient for the equality ordinary symbolic powers ideal having each in some it. characterize naturally unicyclic graphs with unique cycles even their ideals. D^{\prime} obtained from afte...