Hamming weights and Betti numbers of Stanley-Reisner rings associated to matroids

To each linear code over a finite field we associate the matroid of its parity check matrix. We show to what extent one can determine the generalized Hamming weights of the code (or defined for a matroid in general) from various sets of Betti numbers of Stanley-Reisner rings of simplicial complexes associated to the matroid

Polarization and depolarization of monomial ideals with application to multi-state system reliability

Polarization is a powerful technique in algebra which provides combinatorial tools to study algebraic invariants of monomial ideals. We study the reverse of this process, depolarization which leads to a family of ideals which share many common features with the original ideal. Given a squarefree monomial ideal, we describe a combinatorial method to obtain all its depolarizations, and we highlight their similar properties such as the graded Betti numbers. We show that even though they have many similar properties, their differences in dimension make them distinguishable in applications in system reliability theory. In particular, we apply polarization and depolarization tools to study the reliability of multistate coherent systems via binary systems and vice versa. We use depolarization as a tool to reduce the dimension and the number of variables in coherent systems

Algebraic algorithms for the reliability analysis of multi-state k-out-of-n systems

We develop algorithms for the analysis of multi-state k-out-of-n systems and their reliability based on commutative algebra

Betti numbers and minimal free resolutions for multi-state system reliability bounds

Every coherent system has a monomial ideal associated with it and the knowledge of its multigraded Betti numbers provides reliability bounds for the corresponding system, which are the tightest among a certain class of such bounds. Some alternative methods for computing the multigraded Betti numbers are used in this paper and applied in the study of reliability. We obtain special results for well known examples and show that computational commutative algebra techniques can be used beneficially in the reliability analysis of systems of different types

Structure Indicators for Transportation Graph Analysis I: Planar Connected Simple Graphs

The paper deals with the representation of a transportation infrastructure by a planar connected simple graph and aims at studying its features through the analysis of graph properties. All planar and connected graphs with 4 up to 7 edges are analysed and compared to extract the most suitable parameters to investigate some network features. Then, a set of 41 graphs representing some actual underground networks are also analysed. Besides, as a third scenario, the underground network of Milan, along its development in years, is proposed in order to apply the proposed methodology. Many parameters are taken into consideration. Some of them are already discussed in literature, such as the eigenvalues and gaps of adjacency matrix or such as the Bclassical^ parameters α, β, γ. Others, such as the first two Betti numbers, are new for these applications.In order to overcome the problem of comparing features of graphs with different size, the normalisation of these parameters is considered. Some relationships between Betti numbers, eigenvalues, and classical parameters are also investigated. Results show that the eigenvalues and gaps of the adjacency matrix well represent some features of the graphs while combining them with the Betti numbers, a more significant interpretation can be achieved. Particularly, their normalised values are able to describe the increasing complexity of a graph