### Abstract

Suppose that each edge of a connected graph G of order n is independently operational with probability p; the reliability of G is the probability that the operational edges form a spanning connected subgraph. A useful expansion of the reliability is as p^{n-1}∑_{i=0}^{d} H_{i}(1 - p)^{i}, and the Ball-Provan method for bounding reliability relies on Stanley's combinatorial bounds for the H-vectors of shellable complexes. We prove some new bounds here for the H-vectors arising from graphs, and the results here shed light on the problem of characterizing the H-vectors of matroids.

Original language | English (US) |
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Pages (from-to) | 13-36 |

Number of pages | 24 |

Journal | Journal of Algebraic Combinatorics |

Volume | 5 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1996 |

### Keywords

- Graph polynomial
- H-vector
- Matroid
- Network reliability
- Shellable complex

### ASJC Scopus subject areas

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics

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## Cite this

Brown, J. I., & Colbourn, C. J. (1996). Non-Stanley bounds for network reliability.

*Journal of Algebraic Combinatorics*,*5*(1), 13-36. https://doi.org/10.1023/a:1022484229443