Non-Stanley bounds for network reliability

Abstract

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 pn-I y~4do Ill (1 --p)i, and the BalI-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.

Lower Bounds forh-Vectors ofk-CM, Independence, and Broken Circuit Complexes

“…In [6] Brown and Colbourn proved the relative lower bound h r−1 (M ) ≤ rh r (M ) which only involves the rank of M. This can be improved using Theorem 5.4. Theorem 6.6.…”

“…Stanley used the notion of a level ring to establish the relative lower bound h j−i (M ) ≤ h i (M )h j (M ) whenever 0 ≤ i, j ≤ r. In particular, setting j = r, we find that h r−i (M ) ≤ n−r+i−1 i h r (M ). By applying (6) we can obtain similar relative lower bounds for h i−j (M ) in terms of h i (M ) and we can also determine when equality occurs. Proposition 6.5.…”

“…Section 2 contains the basic facts of the short-simplicial h-vector. The main tool for providing relative lower bounds is equation (6). The broken circuit and independence complex of a matroid are described in section 3.…”

The k-fractal of a simplicial complex

“…Studying the roots of polynomials can shed light on their coe cients-and therefore on the underlying combinatorics-but they are also an interesting study in their own right (as has been found in the study of chromatic polynomials [3,9,10,14,15,32,33], reliability polynomials [11,12,38], independence polynomials [13,[16][17][18]21], and others). On the other hand, for various graph-theoretic invariants, there has been considerable interest in investigating limits of the invariant on 'powers' of a graph, where the notion of power corresponds to iterates of some associative binary operation on graphs.…”

“…Formally, if X is a ÿnite set then a nonempty collection C of nonempty subsets of X is a (simplicial) complex if for every set A ∈ C, B ⊂ A implies that B ∈ C as well. The sets X and C contain the vertices and faces, respectively, of the complex, and the largest cardinality of a face is the (combinatorial) dimension of C (this is the deÿnition used most often by combinatorialists (see, for example, [11,12,19]), and is one more than the usual deÿnition of dimension of a complex from algebraic topology). Any matroid (cf.…”