Frank Boesch

Circulants and their connectivities

There is diverse literature on various properties of a class of graphs known as circulants. We present a new result which answers the previously unsolved question of characterizing the connection sequence of circulants having point connectivity equal to point degree. We also develop some theorems regarding a new generalization of connectivity known as super-connectivity. In addition, we give a survey of published results pertinent to the study of connectivity of circulants.

Synthesis of reliable networks - a survey

In contrast to the usual probabilistic model for network reliability, one can use a deterministic model which is called network vulnerability. Many different vulnerability criteria and the related synthesis results are reviewed. These synthesis problems are all graph external questions. Certain reliability synthesis problems can be converted to a vulnerability question. Several open problems and conjectures are presented.

Combinatorial optimization problems in the analysis and design of probabilistic networks

This paper presents some results regarding the design of reliable networks. The problem under consideration involves networks which are undirected graphs having equal and independent edge failure probabilities. The index of reliability is the probability that the network fails (becomes disconnected). For “small” edge failure probabilities and given p and q there exists a class of p vertex, q edge graphs with the property that any graph in the class has a smaller probability of disconnection than any graph outside of the class. We solve the problem of synthesizing graphs in this class.

Graphs as models of communication network vulnerability: Connectivity and persistence

It is well known that the maximum connectivity k of a graph G with p points and q lines is given by [2 q/p]. This is restated in two useful alternative forms which minimize q given p and k, and which maximize p in terms of q and k. We define the persistence of a graph as the smallest number of points whose removal increases the diameter. It is shown that the persistence of a graph of diameter d is the minimum over all pairs of nonadjacent points of the maximum number of disjoint paths of length at most d joining them. A similar result is obtained for line-persistence and it is shown that these invariants are independent of each other.

On the characterization of graphs with maximum number of spanning trees

Abstract A graph G with n nodes and e edges is said to be t-optimal if G has the maximum number of spanning trees among all graphs with the same number of nodes and edges as G . Hitherto, t -optimal graphs have been characterized for the following cases: 1. (a) n = sp , and e = ( s ( s - 1)/2) p 2 , when s and p are positive integers, and s > 1; 2. (b) e ⩽ n + 2; 3. (c) e ⩾ n ( n - 1)/2 - n /2. In this paper we use algebraic techniques involving eigenvalues to determine t -optimal graphs for e ⩾ n ( n - 1)/2 - n + 2. This range is extended to include e = n ( n - 1)/2 - n + 1 and e = n ( n - 1)/2 - n , provided n ( n - 1)/2 - e is a multiple of three.

Line removal algorithms for graphs and their degree lists

An important and basic characterization of a graph is the sequence or list of degrees of that graph. Problems regarding the construction of graphs with specified degrees occur in chemistry and in the design of reliable networks. A list of nonnegative integers is called graphical if there is a graph (called a realization) with the given list as its degree list. The usual algorithms for determining whether a given list is graphical are derived from the effect on a graphical list of the removal of a point from a graph. After reviewing such an algorithm by Havel-Hakimi and its generalization by Wang and Kleitman, we develop a corresponding algorithm based on the removal of a line from a graph. We conclude by reviewing and providing simple proofs of algorithms for a list to be multigraphical due to Hakimi and Butler. The conditions relating a graphical or multigraphical list to the point and line connectivity of their realizations, due to Edmonds, Wang and Kleitman, Boesch and McHugh, and Hakimi, are presented along with new and simple proofs of the multigraph case.

The cohesiveness of a point of a graph

The connectivity contribution or cohesiveness of a point v of graph G is defined as the difference k(G) - k(G - v) where kappa is the usual connectivity symbol. It is shown that if a point v of G has negative cohesiveness, then the set of points adjacent to v is the unique minimum size disconnecting set of G. This theorem has several corollaries including the result that if v has negative cohesiveness in G, then it does not in Γ. Finally we define a cohesiveness triple (n_, n0, n+) of a graph by taking these, respectively, as the number of negative, zero, and positive cohesiveness points of G. The necessary and sufficient conditions for an arbitrary triple to be the cohesiveness triple of a graph are derived.

A coherent model for reliability of multiprocessor networks

Graph G has perfectly reliable nodes and edges that are subject to stochastic failure. The network reliability R is the probability that the surviving edges induce a spanning connected subgraph of G. Analysis problems concern determining efficient algorithms to calculate R, which is known to be NP-hard for general graphs. Synthesis problems concern determining graphs that are, according to some definition, the most reliable in the class of all graphs having a given number of edges and nodes. In applications where the edges are perfectly reliable and the nodes are subject to failure, another measure (residual node connectedness reliability) is defined as the probability that the surviving nodes induce a connected subgraph of G. Referring to such a subset as an operating state, the measure is not coherent because a superset of an operating state need not be an operating state. This paper proposes a new definition of network reliability that handles the case of node failures; it is coherent. We determine many of its properties, and present several analysis and synthesis results.