Ralph Tindell

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.

On the algebraic structure of rooted trees

Figure 1.1 Figure 1.2 These two digraphs, while different, usually represent the same phenomenon, say, the same “computational process.” Our interest in rooted trees stems from the fact that these two digraphs “unfold” into the SAME infinite tree. In some cases at least it is also true that different (i.e. non-isomorphic) trees represent different phenomena (of the same kind). In these cases the unfoldings (i.e. the trees) are surrogates for the phenomena.

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.

The spanning subgraphs of eulerian graphs

It is shown that a connected graph G spans an eulerian graph if and only if G is not spanned by an odd complete bigraph K(2m + 1, 2n + 1). A disconnected graph spans an eulerian graph if and only if it is not the union of the trivial graph with a complete graph of odd order. Exact formulas are obtained for the number of lines which must be added to such graphs in order to get eulerian graphs.

Varieties of ”If-Then-Else“

Four classes of algebras are considered. The algebras in each class contain functions whose behavior models a version of the “if-then-else” instruction. In one version, for example, the algebras contain a function

Connectivity of Cayley Digraphs

\kappa

Algebraic and graph theoretic characterizations of structured flowchart schemes

of four arguments such that

Compatible Orderings on the Metric Theory of Trees

\kappa (x,y,u,v) = u

Chromatic polynomials and network reliability

if

The cohesiveness of a point of a graph

x = y