Algebraic Properties of the Path Ideal of a Tree

Let Γ be a rooted (and directed) tree, and let t be a positive integer. The path ideal I t (Γ ) is generated by monomials that correspond to directed paths of length (t − 1) in Γ . In this paper, we study algebraic properties and invariants of I t (Γ ). We give a recursive formula to compute the graded Betti numbers of I t (Γ ) in terms of path ideals of subtrees. We also give a general bound for the regularity, explicitly compute the linear strand, and investigate when I t (Γ ) has a linear resolution.

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“…In [6] He and Van Tuyl computed the projective dimension of S/I t (L n ). Using (5.1), we can easily recover their formula.…”

Section: 1)mentioning

confidence: 99%

Path ideals of rooted trees and their graded Betti numbers

Bouchat

Hí

O'Keefe

2011

Journal of Combinatorial Theory, Series A

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“…, i n is a path in T k,n . Such ideals are studied in [2,7]. Note that if n is the depth of the tree, then y i0 is the variable corresponding to the root.…”

Section: F Mohammadi E Sáenz-de-cabezón and H P Wynnmentioning

confidence: 99%

The Algebraic Method in Tree Percolation

Mohammadi¹,

Sáenz-de-Cabezón²,

Wynn³

2016

SIAM J. Discrete Math.

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Abstract. We apply the methods of algebraic reliability to the study of percolation on trees. To a complete k-ary tree T k,n of depth n we assign a monomial ideal I k,n on n i=1 k i variables and k n minimal monomial generators. We give explicit recursive formulae for the Betti numbers of I k,n and their Hilbert series, which allow us to study explicitly percolation on T k,n . We study bounds on this percolation and study its asymptotical behavior with the mentioned commutative algebra techniques.Key words. percolation, Betti numbers, monomial ideals, Hilbert series AMS subject classifications. 13D02, 60K35, 05C05 DOI. 10.1137/1510036471. Introduction. The study of monomial ideals has experienced much growth in the last couple of decades, not only from a theoretical point of view [8] but also from the point of view of applications and algorithms [1]. Of particular interest are the relations between the algebra of monomial ideals and the combinatorics of graphs and networks [22,15,21]. In relation with these lines of research, the authors have developed an algebraic theory of system reliability which can be applied to industrial, biological, and communication systems, among others [4,16,19,20]. In this theory, a monomial ideal is associated to a coherent system, and the study of the reliability of the system is performed by studying algebraic invariants of the ideal, such as the Hilbert series and Betti numbers. This algebraic approach to system reliability analysis is an example of enumerative methods for reliability evaluation. In particular, it is an improvement of the inclusion-exclusion method, which is the most general one for coherent systems [4,16].A main difficulty and the first step in the use of monomial ideals to study the reliability of coherent systems is the enumeration of the working and failure states of the system. This made the authors focus on several widely used and structured systems, like k-out-of-n systems [16], series-parallel systems [18], all-terminal networks [13,14,12], and the more general category of two-terminal networks [17,12]. The present paper follows this line extending the application of the algebraic approach to reliability analysis to a more general situation, which allows us to introduce these techniques in percolation theory, a branch of probability theory.In the setting of two-terminal networks the situation is the following. Consider a network as a simple connected graph G = (V, E), where V is the set of vertices (nodes) and E is the set of edges (connections). To have a two-terminal network, we select two special vertices in the graph, s (source) and t (target), and study the connections between s and t in the network. We consider that vertices are reliable but

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“…Path ideals were first introduced by Conca and De Negri [9]. Further properties have been developed by He and Van Tuyl [35], Bouchat, Hà, and O'Keefe [3], and Alilooee and Faridi [1]. Construction 1.41 (Edge and cover ideals of hypergraphs).…”

Section: Construction 140 (Path Ideals)mentioning

confidence: 99%