Related papers: Regularity of Binomial Edge Ideals of Certain Bloc…
The v-number of a graded ideal is an invariant recently introduced in the context of coding theory, particularly in the study of Reed--Muller-type codes. In this work, we study the localized v-numbers of a binomial edge ideal $J_G$…
We present two new problems on lower bounds for resolution Betti numbers of monomial ideals generated in a fixed degree. The first concerns any such ideal and bounds the total Betti numbers, while the second concerns ideals that are…
We introduce binomial edge ideals attached to a simple graph $G$ and study their algebraic properties. We characterize those graphs for which the quadratic generators form a Gr\"obner basis in a lexicographic order induced by a vertex…
Let $G$ be a finite simple graph on $n$ vertices and $J_G$ denote the corresponding binomial edge ideal in the polynomial ring $S = K[x_1, \ldots, x_n, y_1, \ldots, y_n].$ In this article, we compute the Hilbert series of binomial edge…
We study the problem of enumerating Braess edges for Kemeny's constant in trees. We obtain bounds and asympotic results for the number of Braess edges in some families of trees.
In this paper we provide a full combinatorial characterization of sequentially Cohen-Macaulay binomial edge ideals of closed graphs. In addition, we show that a binomial edge ideal of a closed graph is approximately Cohen-Macaulay if and…
This work establishes combinatorial bounds on the Castelnuovo-Mumford regularity of edge ideals for trees and their multi-whiskered variants. For a tree \( T \), we give bounds for the Castelnuovo-Mumford regularity of \( I(T) \) in terms…
Let $G$ be a finite simple graph on $n$ non-isolated vertices, and let $J_G$ be its binomial edge ideal. We determine almost all pairs $(\text{projdim}(J_G),\text{reg}(J_G))$, where $G$ ranges over all finite simple graphs on $n$…
There is a natural infinite graph whose vertices are the monomial ideals in a polynomial ring. The definition involves Gr\"obner bases or the action of an algebraic torus. We present algorithms for computing the (affine schemes…
We obtain sharp lower and upper bounds for the number of maximal (under inclusion) independent sets in trees with fixed number of vertices and diameter. All extremal trees are described up to isomorphism.
In this article, we study algebraic properties of binomial edge ideals of Levi graphs associated with certain plane curve arrangements. Using combinatorial properties of Levi graphs, we discuss the Cohen-Macaulayness of binomial edge ideals…
We study the symbolic $F$-splitness of families of binomial edge ideals. We also study the strong $F$-regularity of the symbolic blowup algebras of families of binomial edge ideals. We make use of Fedder-like criteria and combinatorial…
We determine the arithmetical rank of every edge ideal of a Ferrers graph.
We prove polynomial upper and lower bounds on the expected size of the maximum agreement subtree of two random binary phylogenetic trees under both the uniform distribution and Yule-Harding distribution. This positively answers a question…
This paper gives exact formulas for the regularity of powers of edge ideals of an edge-weighted integrally closed cycle.
In this paper, we describe primary decomposition of the edge ideal of the join of some graphs in terms of that information of the edge ideal of every weighted oriented graph. Meanwhile, we also study depth and regularity of symbolic powers…
We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature…
We determine the sharp threshold for the containment of all $n$-vertex trees of bounded degree in random geometric graphs with $n$ vertices. This provides a geometric counterpart of Montgomery's threshold result for binomial random graphs,…
In this article, we prove that for several classes of graphs, the Castelnuovo-Mumford regularity of symbolic powers of their edge ideals coincide with that of their ordinary powers.
Let $G$ be a graph and $I(G)$ its edge ideal. In this paper, we give a complete characterization of the graphs $G$ for which $\reg(R/I(G)) = 3$.