Related papers: Forests whose matching powers are linear
In this paper, we give a complete description of the associated primes of each power of the edge ideal of an increasing weighted tree.
For any integer $n$, we classify all trees whose $n$-path ideals have linear quotients.
We show that for the edge ideals of a certain class of forests, the arithmetical rank equals the projective dimension.
In this paper we provide some exact formulas for projective dimension and the regularity of powers of edge ideals of vertex-weighted rooted forests. These formulas are functions of the weight of the vertices and the number of edges. We also…
This paper gives exact formulas for the regularity of edge ideals of edge-weighted integrally closed trees. In addition, we provide some linear upper bounds on the regularity of powers of such ideals.
In weighted trees, all edges are endowed with positive integral weight. We enumerate weighted bicolored plane trees according to their weight and number of edges.
Let $I(G)^{[k]}$ denote the $k$th squarefree power of the edge ideal of $G$. When $G$ is a forest, we provide a sharp upper bound for the regularity of $I(G)^{[k]}$ in terms of the $k$-admissable matching number of $G$. For any positive…
Elementary arguments show that a tree or forest is determined (up to isomorphism) by binary matroids defined using the adjacency matrix.
We give criteria for graded ideals to have the property that all their powers are componentwise linear. Typical examples to which our criteria can be applied include the vertex cover ideals of certain finite graphs.
We introduce the concept of matching powers of monomial ideals. Let $I$ be a monomial ideal of $S=K[x_1,\dots,x_n]$, with $K$ a field. The $k$th matching power of $I$ is the monomial ideal $I^{[k]}$ generated by the products $u_1\cdots u_k$…
We characterize the extremal trees that maximize the number of almost-perfect matchings, which are matchings covering all but one or two vertices, and those that maximize the number of strong almost-perfect matchings, which are matchings…
We describe all the trees with the property that the corresponding edge ideal of the square of the tree has a linear resolution. As a consequence, we give a complete characterization of those trees $T$ for which the square is co-chordal,…
In this paper we consider graded ideals in a polynomial ring over a field and ask when such an ideal has the property that all of its powers have a linear resolution. In particular it is shown that all powers of a monomial ideal with…
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We classify all convex polyomino ideals which are linearly related or have a linear resolution. Convex stack polyominoes whose ideals are extremal Gorenstein are also classified. In addition, we characterize, in combinatorial terms, the…
Given a tree of weighted vertices, it is sometimes possible to break the tree into two equally-weighted subtrees within an allowable error. We give a fast algorithm that finds an edge which breaks the tree into equal-weight components or…
Trees or rooted trees have been generously studied in the literature. A forest is a set of trees or rooted trees. Here we give recurrence relations between the number of some kind of rooted forest with $k$ roots and that with $k+1$ roots on…
The matrices of spanning rooted forests are studied as a tool for analysing the structure of networks and measuring their properties. The problems of revealing the basic bicomponents, measuring vertex proximity, and ranking from preference…
By weighted tree we understand such connected tree,that: a) each its vertex and each edge have a positive integer weight; b) the weight of each vertex is equal to the sum of weights of outgoing edges. Each tree has a binary structure --- we…
In this paper we provide some exact formulas for the projective dimension and the regularity of edge ideals associated to vertex weighted rooted forests and oriented cycles. As some consequences, we give some exact formulas for the depth of…