Let $I(G)^{[k]}$ denote the $k$th squarefree power of edge ideal $G$. When $G$ is a forest, we provide sharp upper bound for regularity in terms $k$-admissable matching number For any positive integer $k$, classify all forests such that has linear resolution. We also give combinatorial formula $I(G)^{[2]}$ forest

Let G be a finite simple graph on the vertex set $$V(G) = \{x_{1}, \ldots , x_{n}\}$$ and match(G), min-match(G) ind-match(G) matching number, minimum number induced of G, respectively. $$K[V(G)] K[x_{1}, x_{n}]$$ denote polynomial ring over field K $$I(G) \subset K[V(G)]$$ edge ideal G. The relationship between these graph-theoretic invariants ring-theoretic quotient K[V(G)]/I(G) has been stud...

We study the regularity of small symbolic powers and integral closure edge ideals. also prove that ideals graphs with at most two odd cycles is same as their powers.

We provide the regularity and Cohen–Macaulay type of binomial edge ideals cones, we show extremal Betti numbers some classes ideals: bipartite fan graphs. In addition, compute Hilbert–Poincaré series

 The rings considered in this article are  commutative  with identity which admit at least two  nonzero annihilating ideals. Let $R$ be a ring. Let $mathbb{A}(R)$ denote the set of all annihilating ideals of $R$ and let $mathbb{A}(R)^{*} = mathbb{A}(R)backslash {(0)}$. The annihilating-ideal graph of $R$, denoted by $mathbb{AG}(R)$  is an undirected simple graph whose vertex set is $mathbb{A}(R...

Let C be a uniform clutter and let I = I(C) be its edge ideal. We prove that if C satisfies the packing property (resp. max-flow min-cut property), then there is a uniform Cohen-Macaulay clutter C1 satisfying the packing property (resp. max-flow min-cut property) such that C is a minor of C1. For arbitrary edge ideals of clutters we prove that the normality property is closed under parallelizat...

The well known correspondence between even cycles of an undirected graph and polynomials in a binomial ideal associated to a graph is extended to odd cycles and polynomials in another binomial ideal. Other binomial ideals associated to an undirected graph are also introduced. The results about them with topics on monomial ideals are used in order to show decision procedures for bipartite graphs...

Given a simple graph G on n vertices, we prove that it is possible to reconstruct several algebraic properties of the edge ideal from the deck of G, that is, from the collection of subgraphs obtained by removing a vertex from G. These properties include the Krull dimension, the Hilbert function, and all the graded Betti numbers i,j where j <n. We also state many further questions that arise fro...