Cohen-Macaulay rings in network reliability

Jason I. Brown, Charles Colbourn, David G. Wagner

Research output: Contribution to journalArticle

5 Citations (Scopus)

Abstract

For any simplicial complex Δ and field K, one can associate a graded K-algebra K[Δ] (the Stanley-Reisner ring). For certain Δ and K, the Stanley-Reisner rings have a homogeneous system of parameters, Θ, such that K[Δ]/〈Θ〉 is finite-dimensional, and coefficients of its Hilbert series are the h-vector of Δ. The previous constructions of Θ were noncombinatorial. In the special case of cographic matroids, we give (for any field K) a combinatorial description of a homogeneous system of parameters in terms of the graph structure, as well as an explicit basis for the resulting quotient algebra. The results have applications to a central problem of reliability, namely the association of a multicomplex to a connected graph, such that the reliability is a simple function of the rank numbers.

Original languageEnglish (US)
Pages (from-to)377-392
Number of pages16
JournalSIAM Journal on Discrete Mathematics
Volume9
Issue number3
StatePublished - Aug 1996
Externally publishedYes

Fingerprint

Stanley-Reisner Ring
Cohen-Macaulay Ring
Network Reliability
Algebra
H-vector
Hilbert Series
Simplicial Complex
Matroid
Connected graph
Quotient
Coefficient
Graph in graph theory

Keywords

  • Cohen-Macaulay ring
  • Graph
  • Gröbner basis
  • Homogeneous system of parameters
  • Reliability

ASJC Scopus subject areas

  • Applied Mathematics
  • Discrete Mathematics and Combinatorics

Cite this

Cohen-Macaulay rings in network reliability. / Brown, Jason I.; Colbourn, Charles; Wagner, David G.

In: SIAM Journal on Discrete Mathematics, Vol. 9, No. 3, 08.1996, p. 377-392.

Research output: Contribution to journalArticle

Brown, JI, Colbourn, C & Wagner, DG 1996, 'Cohen-Macaulay rings in network reliability', SIAM Journal on Discrete Mathematics, vol. 9, no. 3, pp. 377-392.
Brown, Jason I. ; Colbourn, Charles ; Wagner, David G. / Cohen-Macaulay rings in network reliability. In: SIAM Journal on Discrete Mathematics. 1996 ; Vol. 9, No. 3. pp. 377-392.
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