### 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 language | English (US) |
---|---|

Pages (from-to) | 377-392 |

Number of pages | 16 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 9 |

Issue number | 3 |

State | Published - Aug 1996 |

Externally published | Yes |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Applied Mathematics
- Discrete Mathematics and Combinatorics

### Cite this

*SIAM Journal on Discrete Mathematics*,

*9*(3), 377-392.

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

Research output: Contribution to journal › Article

*SIAM Journal on Discrete Mathematics*, vol. 9, no. 3, pp. 377-392.

}

TY - JOUR

T1 - Cohen-Macaulay rings in network reliability

AU - Brown, Jason I.

AU - Colbourn, Charles

AU - Wagner, David G.

PY - 1996/8

Y1 - 1996/8

N2 - 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.

AB - 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.

KW - Cohen-Macaulay ring

KW - Graph

KW - Gröbner basis

KW - Homogeneous system of parameters

KW - Reliability

UR - http://www.scopus.com/inward/record.url?scp=0030367185&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0030367185&partnerID=8YFLogxK

M3 - Article

VL - 9

SP - 377

EP - 392

JO - SIAM Journal on Discrete Mathematics

JF - SIAM Journal on Discrete Mathematics

SN - 0895-4801

IS - 3

ER -