Related papers: Depth of the 3-path Ideal of Square of a Path

We consider the closed neighborhood ideal of square of the path graph and study some of its algebraic and homological invariants. We compute the height, the projective dimension and the Castelnuovo-Mumford regularity. We prove that these…

We compute the depth and Stanley depth for the quotient ring of the path ideal of length $3$ associated to a $n$-cyclic graph, given some precise formulas for depth when $n\not\equiv 1\,(\mbox{mod}\ 4)$, tight bounds when $n\equiv…

We compute the projective dimension and regularity of $3$-path ideals of arbitrary graphs with at most one cycle.

Given a tree T on n vertices, there is an associated ideal I of a polynomial ring in n variables over a field, generated by all paths of a fixed length of T. We show that such an ideal always satisfies the Konig property and classify all…

Let $G=P_n$ be a path graph with cover ideal $J(P_n)$. By using Hochster's depth formula, we prove the explicit formulae to compute the depth functions of powers of cover ideals of paths.

By generalizing the notion of the path ideal of a graph, we study some algebraic properties of some path ideals associated to a line graph. We show that the quotient ring of these ideals are always sequentially Cohen-Macaulay and also…

We provide exact formulas for the depth of the quotient ring of powers of the edge ideal of an increasing weighted path.

Let $I_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\; x_{n-m+1}\cdots x_n)$ be the $m$-path ideal of the path graph of length $n$, in the ring $S=K[x_1,\ldots,x_n]$. We prove that: $$\mathtt{depth}(S/I_{n,m}^t)=\begin{cases}…

Let $J_{n,m}:=(x_1x_2\cdots x_m,\; x_2x_3\cdots x_{m+1},\; \ldots,\; x_{n-m+1}\cdots x_n,\; x_{n-m+2}\cdots x_nx_1, \ldots, x_nx_1\cdots x_{m-1})$ be the $m$-path ideal of the cycle graph of length $n$, in the ring of polynomials…

We prove that the set of all paths of a fixed length in a complete multipartite graph is the bases of a matroid. Moreover, we discuss the Cohen-Macaulayness and depth of powers of $t$-path ideals of a complete multipartite graph.

In this article, we give combinatorial formulas for the regularity and the projective dimension of $3$-path ideals of chordal graphs, extending the well-known formulas for the edge ideals of chordal graphs given in terms of the induced…

We compute the depth of powers of edge ideals of integrally closed edge-weighted paths.

The aim of this paper is to give a formula for the Stanley depth of quotient of powers of the path ideal. As a consequence, we establish that the behaivior of the Stanley depth of quotient of powers of the path ideal is the same as a…

Let $I$ be the edge ideal of a cycle of length $n \ge 5$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_n]$. We prove that for $2 \le t < \lceil (n+1)/2 \rceil$, $$\operatorname{depth} (S/I^t) = \lceil \frac{n -t + 1}{3} \rceil.$$ When…

Let $I$ be the edge ideal of a Cohen-Macaulay tree of dimension $d$ over a polynomial ring $S = \mathrm{k}[x_1,\ldots,x_{d},y_1,\ldots,y_d]$. We prove that for all $t \ge 1$, $$\operatorname{depth} (S/I^t) = \operatorname{max} \{d -t + 1, 1…

We give tight bounds for the Stanley depth of the quotient ring of the path ideal of a cycle graph. In particular, we prove that it satisfies the Stanley inequality.

We consider the path ideal associated to a line graph, we compute \texttt{sdepth} for its quotient ring and note that it is equal with its \texttt{depth}. In particular, it satisfies the Stanley inequality.

Let $G$ be a graph and let $I$ be the edge ideal of $G$. Our main results in this article provide lower bounds for the depth of the first three powers of $I$ in terms of the diameter of $G$. More precisely, we show that $\depth R/I^t \geq…

We consider a class of graphs $G$ such that the height of the edge ideal $I(G)$ is half of the number $\sharp V(G)$ of the vertices. We give Cohen-Macaulay criteria for such graphs.

Let $k$ be a positive integer. We compute depth and Stanley depth of the quotient ring of the edge ideal associated to the $k^{th}$ power of a path on $n$ vertices. We show that both depth and Stanley depth have the same values and can be…