Jack E. Graver

The groups of the generalized Petersen graphs

1. Introduction . For integers n and k with 2 ≤ 2k n , the generalized Petersen graph G(n, k) has been defined in (8) to have vertex-set and edge-set E(G(n, k)) to consist of all edges of the form where i is an integer. All subscripts in this paper are to be read modulo n , where the particular value of n will be clear from the context. Thus G(n, k) is always a trivalent graph of order 2 n , and G (5, 2) is the well known Petersen graph. (The subclass of these graphs with n and k relatively prime was first considered by Coxeter ((2), p. 417ff.).)

Mean distance in a graph

Abstract The average or mean of the distances between vertices in a connected graph Γ, μ(Γ), is a natural measure of the compactness of the graph. In this paper we compute bounds for μ(Γ) in terms of the number of vertices in Γ and the diameter of Γ. We prove a formula for computing μ(Γ) when Γ is a tree which is particularly useful when Γ has a high degree of symmetry. Finally, we present algorithms for μ(Γ) which are amenable to computer implementation.

Some graph theoretic results associated with Ramsey's theorem

We consider the numbers associated with Ramseys theorem as it pertains to partitions of the pairs of elements of a set into two classes. Our purpose is to give a unified development of enumerative techniques which give sharp upper bounds on these numbers and to give constructive methods for partitions to determine lower bounds on these numbers. Explicit computations include the values of R(3, 6) and R(3, 7) among others. Our computational techniques yield the upper bound R(x, y)≤cyx−1log log y/logy for x≥3.

The module structure of integral designs

Abstract The existence problem for t-designs with prescribed parameters is solved by allowing positive and negative integral multiplicities for the blocks.

On the foundations of linear and integer linear programming I

In this paper we consider the question: how does the flow algorithm and the simplex algorithm work? The usual answer has two parts: first a description of the “improvement process”, and second a proof that if no further improvement can be made by this process, an optimal vector has been found. This second part is usually based on duality -a technique not available in the case of an arbitrary integer programming problem. We wish to give a general description of “improvement processes” which will include both the simplex and flow algorithms, which will be applicable to arbitrary integer programming problems, and which will inthemselves assure convergence to a solution.Geometrically both the simplex algorithm and the flow algorithm may be described as follows. At the ith stage, we have a vertex (or feasible flow) to which is associated a finite set of vectors, namely the set of edges leaving that vertex (or the set of unsaturated paths). The algorithm proceeds by searching among this special set for a vector along which the gain function is increasing. If such a vector is found, the algorithm continues by moving along this vector as far as is possible while still remaining feasible. The search is then repeated at this new feasible point.We give a precise definition for sets of vectors, called test sets, which will include those sets described above arising in the simplex and flow algorithms. We will then prove that any “improvement process” which searches through a test set at each stage converges to an optimal point in a finite number of steps. We also construct specific test sets which are the natural extensions of the test sets employed by the flow algorithm to arbitrary linear and integer linear programming problems.

Kekulé structures and the face independence number of a fullerene

We explore the relationship between Kekule structures and maximum face independence sets in fullerenes: plane trivalent graphs with pentagonal and hexagonal faces. For the class of leap-frog fullerenes, we show that a maximum face independence set corresponds to a Kekule structure with a maximum number of benzene rings and may be constructed by partitioning the pentagonal faces into pairs and 3-coloring the faces with the exception of a very few faces along paths joining paired pentagons. We also obtain some partial results for non-leap-frog fullerenes.

The independence numbers of fullerenes and benzenoids

We explore the structure of the maximum vertex independence sets in fullerenes: plane trivalent graphs with pentagonal and hexagonal faces. At the same time, we will consider benzenoids: plane graphs with hexagonal faces and one large outer face. In the case of fullerenes, a maximum vertex independence set may constructed as follows: (i) Pair up the pentagonal faces. (ii) Delete the edges of a shortest path in the dual joining the paired faces to get a bipartite subgraph of the fullerene. (iii) Each of the deleted edges will join two vertices in the same cell of the bipartition; eliminating one endpoint of each of the deleted edges results in two independent subsets.The main part of this paper is devoted to showing that for a properly chosen pairing, the larger of these two independent subsets will be a maximum independent set. We also prove that the construction of a maximum vertex independence set in a benzenoid is similar with the dual paths between pentagonal faces replaced by dual circuits through the outside face. At the end of the paper, we illustrate this method by computing the independence number for each of the icosahedral fullerenes.

Boolean designs and self-dual matroids☆

Abstract In this paper we consider a variety of questions in the context of Boolean designs. For example, Erdos asked: How many subsets of an n-set can be found so that pairwise their intersections are all even (odd)? E. Berlekamp [2] and the author both answered this question; the answer is approximately 2[ 1 2 n]. Another question which can be formulated in terms of Boolean designs was asked by J. A. Bondy and D. J. A. Welsh [1]. For what values of d can one find a connected binary matroid of rank d which is identically self-dual? We prove that such matroids exist for all d except 2, 3, and 5. The paper ends with a discussion of more general modular designs and with constructions of some identically self-dual matroids representable over the field of three elements.

Mean distance in a directed graph

In this paper we study the mean of the distances in a directed graph and compare the results obtained with the corresponding results for undirected graphs. We derive a formula which enables the computation of best possible upper and lower bounds for the mean of the distances in a directed graph. Finally we present a table of values of the mean distance for small graphs and directed graphs.

You May Rely on the Reliability Polynomial for Much More Than You Might Think

ABSTRACT The reliability polynomial R 𝒮(p) of a collection 𝒮 of subsets of a finite set X has been extensively studied in the context of network theory. There, X is the edge set of a graph (V, X) and 𝒮 the collection of the edge sets of certain subgraphs. For example, we may take 𝒮 to be the collection of edge sets of spanning trees. In that case, R 𝒮(p) is the probability that, when each edge is included with the probability p, the resulting subgraph is connected. The second author defined R 𝒮(p) in an entirely different way enabling one to glean additional information about the collection 𝒮 from R 𝒮(p). Illustrating the extended information available in the reliability polynomial is the main focus of this article while demonstrating the equivalence of these two definitions is the main theoretical result.