Edgar M. Palmer

On acyclic simplicial complexes

The higher dimensional concepts corresponding to trees are developed and studied. In order to enumerate these 2-dimensional structures called 2-trees, a dissimilarity characteristic theory is investigated. By an appropriate application of certain combinatorial techniques, generating functions are obtained for the number of 2-trees. These are specialized to count those 2-trees embeddable in the plane, thus providing a new approach to the old problem of determining the number of triangulations of a polygon.

On the spanning tree packing number of a graph: a survey

Abstract The spanning tree packing number or STP number of a graph G is the maximum number of edge-disjoint spanning trees contained in G . We use an observation of Paul Catlin to investigate the STP numbers of several families of graphs including quasi-random graphs, regular graphs, complete bipartite graphs, cartesian products and the hypercubes.

On the cell-growth problem for arbitrary polygons

We solve a variation of the cell-growth problem by enumerating (unlabeled) polygonal clusters, whose constituent cells are all n-gons. The corresponding labeled problem had already been solved by one of us and its solution provides an initial step in the procedure developed here. It will be seen that when n = 3, this amounts to counting triangulations of the disk.

Enumeration of finite automata

Harary ( 1960, 1964), in a survey of 27 unsolved problems in graphical enumeration, asked for the number of different finite automata. Recently, Harrison (1965) solved this problem, but without considering automata with initial and final states. With the aid of the Power Group Enumeration Theorem (Harary and Palmer, 1965, 1966) the entire problem can be handled routinely. The method involves a confrontation of several different operations on permutation groups. To set the stage, we enumerate ordered pairs of functions with respect to the product of two power groups. Finite automata are then concisely defined as certain ordered pahs of functions. We review the enumeration of automata in the natural setting of the power group, and then extend this result to enumerate automata with initial and terminal states.

Balancing the n -Cube: A Census of Colorings

Weights of 1 or 0 are assigned to the vertices of the n-cube in n-dimensional Euclidean space. Such an n-cube is called balanced if its center of mass coincides precisely with its geometric center. The seldom-used n-variable form of Pólyas enumeration theorem is applied to express the number Nn, 2k of balanced configurations with 2k vertices of weight 1 in terms of certain partitions of 2k. A system of linear equations of Vandermonde type is obtained, from which recurrence relations are derived which are computationally efficient for fixed k. It is shown how the numbers Nn, 2k depend on the numbers An, 2k of specially restricted configurations. A table of values of Nn, 2k and An, 2k is provided for n = 3, 4, 5, and 6. The case in which arbitrary, nonnegative, integral weights are allowed is also treated. Finally, alternative derivations of the main results are developed from the perspective of superposition.

Enumeration of graphs with signed points and lines

Our object is to enumerate graphs in which the points or lines or both are assigned positive or negative signs. We also treat several associated problems for which these configurations are self-dual with respect to sign change. We find that the solutions to all of these counting problems can be expressed as special cases of one general formula involving the concatenation of the cycle index of the symmetric group with that of its pair group. This counting technique is based on Polyas Enumeration Theorem and the Power Group Enumeration Theorem. Using a suitable computer program, we list the number of graphs of each type considered up to twelve points. Sharp asymptotic estimates are also obtained.

On the Number of Trees in a Random Forest

The analytic methods of Polya, as reported in [l, 61 are used to determine the asymptotic behavior of the expected number of (unlabeled) trees in a random forest of orderp. Our results can be expressed in terms of q = .338321856899208.. ., the radius of convergence of t(x) which is the ordinary generating function for trees. We have found that the expected number of trees in a random forest approaches 1 + xFz1 t(~~) = 1.755510... and the form of this result is the same for other species of trees. The problem of estimating the number of trees in a large, random labeled forest was treated in Moon’s book Counting Labeled Trees [3, p. 291. It was found that the average number of labeled trees in all labeled forests ofp points approaches 3/2 as a limit as p increases. We have investigated the same question for unlabeled trees and have found that in this case the average number of trees also approaches a constant, namely 1.755510**. This average an be expressed in terms of the ordinary generating function t(x) for trees and its radius of convergence 7. We use the notation and terminology of the book GraphicaZ Enumeration [I] and the analytic methods of Polya as reported in [ 1, 61. Let F,, be the number of forests of order

The number of ways to label a structure

It has been observed that the number of different ways in which a graph withp points can be labelled isp! divided by the number of symmetries, and that this holds regardless of the species of structure at hand. In this note, a simple group-theoretic proof is provided.

Counting Claw-Free Cubic Graphs

Let Hn be the number of claw-free cubic graphs on 2n labeled nodes. Combinatorial reductions are used to derive a second order, linear homogeneous differential equation with polynomial coefficients whose power series solution is the exponential generating function for Hn. This leads to a recurrence relation for Hn which shows Hn to be P-recursive and which enables the sequence to be computed efficiently. Thus the enumeration of labeled claw-free cubic graphs can be added to the handful of known counting problems for regular graphs with restrictions which have been proved P-recursive.

Enumeration of self-converse digraphs

How many digraphs are isomorphic with their own converses? Our object is to derive a formula for the counting polynomial d p ′(x) which has as the coefficient of x q , the number of “self-converse” digraphs with p points and q lines. Such a digraph D has the property that its converse digraph D′ (obtained from D by reversing the orientation of all lines) is isomorphic to D . The derivation uses the classical enumeration theorem of Polya [9[ as applied to a restriction of the power group [6] wherein the permutations act only on 1–1 functions.