Francis T. Boesch

On unreliability polynomials and graph connectivity in reliable network synthesis

The analysis and synthesis of reliable large-scale networks typically involve a graph theoretic model. We give a survey of the graph theoretic notions which are relevant to the synthesis problem. It is shown how a number of unsolved graph extremal problems relate to the synthesis question.

Spanning tree formulas and chebyshev polynomials

AbstractThe Kirchhoff Matrix Tree Theorem provides an efficient algorithm for determiningt(G), the number of spanning trees of any graphG, in terms of a determinant. However for many special classes of graphs, one can avoid the evaluation of a determinant, as there are simple, explicit formulas that give the value oft(G). In this work we show that many of these formulas can be simply derived from known properties of Chebyshev polynomials. This is demonstrated for wheels, fans, ladders, Moebius ladders, and squares of cycles. The method is then used to derive a new spanning tree formula for the complete prismRn(m) =Km ×Cn. It is shown that

Least reliable networks and the reliability domination

2^{\left( {\begin{array}{*{20}c} n \\ 2 \\ \end{array} } \right)\left( {1 - \frac{1}{{r - 1}} + o\left( 1 \right)} \right)}

Realizability of p-point graphs with prescribed minimum degree, maximum degree, and point-connectivity

whereTn(x) is thenth order Chebyshev polynomial of the first kind.

Realizability of p-point, q-line graphs with prescribed point connectivity, line connectivity, or minimum degree

A well-known model for network reliability studies consists of an undirected graph with perfectly reliable nodes and equal and independent edge failure probabilities. The measure of reliability is then defined to be the probability that the graph is connected. A well-defined synthesis problem is to find the graph that minimizes the failure probability given the number of nodes n, the number of edges e, and the edge failure rate ρ. In this work, we consider the possibility of the existence of a fixed graph that is optimal for all possible ρ. It is simple to verify that such graphs exist when e = n − 1, or n. Herein, we show that they also exist when e = n + 1 and n + 2.

An Edge Extremal Result for Subcohesion

A well-known model in communication network reliability consists of an undirected graph G whose edges operate independently with the same probability p. Then the reliability, R(G,p) of G, is the probability that G is connected. It is known that R(G,p) is a polynomial in p and its coefficient of the least-order term is the number of spanning trees t(G), while the coefficient of the highest-order term is the reliability domination d(G) of G. Presented is a complete characterization of graphs that achieve the minimum absolute value mod d(G) mod over the class of n-node, e-edge connected graphs. Furthermore, the class of graphs that yield minimum t(G) is shown to minimize mod d(G) mod . The results have applications in the synthesis of least-reliable networks. >

Graphs with most number of three point induced connected subgraphs

Abstract : An important problem in reliability theory is to determine the reliability of a system from the reliability of its components. If E is a finite set of components, then certain subsets of E are prescribed to be the operating states of the system. A formation is any collection F of minimal operating states whose union is E. Reliability domination is defined as the total number of odd cardinality formations minus the total number of even cardinality formations. The purpose of this paper is to establish some new results concerning reliability domination. In the special case where the system can be identified with a graph or digraph, these new results lead to some new graph- theoretic properties and to simple proofs of certain known theorems. The pertinent graph-theoretic properties include spanning trees, acyclic orientations, Whitneys broken cycles, and Tuttes internal activity associated with the chromatic polynomial.

Network reliability - the state of the art foreword

Abstract In a recent paper, we gave a generalization of extremal problems involving certain graph-theoretic invariants. In that work, we defined a ( p , Δ, δ, λ) graph as a graph having p points, maximum degree Δ, minimum degree δ, and line-connectivity λ. An arbitrary quadruple of integers ( a, b, c, d ) was called ( p , Δ, δ, λ) realizable if there is a ( p , Δ, δ, λ) graph with p = a , Δ = b , δ = c , and λ = d . In this work, we consider the more difficult case of ( p , Δ, δ, κ) realizability, where κ is the point-connectivity. Necessary and sufficient conditions for a quadruple to be ( p , Δ, δ, κ) realizable are derived.