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
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 journal › Article
}
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 -