### 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) |
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Pages (from-to) | 377-392 |

Number of pages | 16 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 9 |

Issue number | 3 |

DOIs | |

State | Published - Aug 1996 |

Externally published | Yes |

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### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*SIAM Journal on Discrete Mathematics*,

*9*(3), 377-392. https://doi.org/10.1137/S0895480194270780