Daniel Gross

Uniformly optimally reliable graphs

A graph with n nodes and e edges, where the nodes are perfectly reliable and the edges fail independently with equal probability ρ, is said to be uniformly optimally reliable (UOR) if it has the greatest reliability among all graphs with the same number of nodes and edges for all values of ρ. UOR simple graphs have been identified in the classes e = n − 1, e = n, e = n + 1, and e = n + 2 (Boesch et al., Networks21 (199) 181–194). In this paper, we demonstrate that the UOR simple graphs in these classes are also UOR when the classes are extended to include multigraphs.

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.

Spanning tree results for graphs and multigraphs : a matrix-theoretic approach

Graph Theory Background Matrix Theory Background, including Kroenecker Products, and Proofs of the Binet - Cauchy and Courant - Fischer Theorems Spanning Tree Results for a Host of Graphs as well as Multigraphs Node-Arc Incidence Matrix Temperleys B Matrix. Multigraphs Eigenvalues and Eigenvalue Bounds A Lagrange Multiplier Approach to the Spanning Tree Problem.

On the fundamental group of the fixed points of an uhipotent action

We show that for a meromorphic action of an unipotent linear algebraic group over on a compact connected reduced complex space, the fundamental group of the fixed point set of the action surjects onto the fundamental group of the space.