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 P'olya's enumeration theorem is applied to express the number N n;2k of balanced configurations with 2k vertices of 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 N n;2k depend on the numbers A n;2k of specially restricted configurations. A table of values of N n;2k and A n;2k is provided for n = 3; 4; 5 and 6. The case in which arbitrary, non-negative, integral weights are allowed is also treated. Finally, alternative derivations of the main results are developed from the perspective of superposition. Key words. n-cube, boolean functions, P'olya enumeration, supe...

This paper consists of two related parts. In the first part the theory of D-finite power series in several variables and the theory of symmetric functions are used to prove P-recursiveness for regular graphs and digraphs and related objects, that is, that their counting sequences satisfy linear homogeneous recurrences with polynomial coefficients. Previously this has been accomplished only for small degrees. See, for example, GOULDEN, JACKSON, and REILLY [7], GOULDEN and JACKSON [6], and READ [16, 18]. These authors found the recurrences satisfied by the sequences in question. Although the methods used here are in principle constructive, we are concerned here only with the question of existence of these recurrences and we do not find them. In the second part we consider a generalization of symmetric functions in several sets of variables, first studied by MACMAHON [13 ; 14, Vol. 2, pp. 280--326]. MacMahon's generalized symmetric functions can be used to find explicit formulas and prove P-recursiveness for some objects to which the theory of ordinary symmetric functions does not apply, such as Latin rectangles and 0-1 matrices with zeros on the diagonal and given row and column sums.

The aim is a description of discrete mathematics used in a project devoted to the implementation of a software package for the simulation of combinatorial chemistry.

Let g(n) denote the number of unlabeled graphs on n nodes, and let e(n) denote its 2-part, i.e., the exponent of the largest power of 2 which divides g(n). It is shown that for odd n 5, e(n) = (n + 1)=2 \Gamma blog 2 nc and for even n 4 e(n) n=2 \Gamma blog 2 nc with equality if, and only if, n is a power of 2. Similarly, let t(n) denote the number of unlabeled tournaments on n nodes and r(n) its 2-part. It is shown that for all odd n, r(n) = (n \Gamma 1)=2 and for all even n 4 r(n) n=2 with equality if, and only if, '(n)=2 is odd. Here '(n) is the Euler totient function. A preliminary version of this paper (without tournament results) was presented at the 22nd Southeastern International Conference on Combinatorics, Graph Theory, and Computing in Baton Rouge, LA, on February 11, 1991. The present version appears in the proceedings of that conference, Congr. Numer. 82 (1991) 139-155. 1 Introduction Let g(n) be the number of nonisomorphic graphs on n nodes, often ref...

. During the last decade, a lot of progress has been made in the enumerative branch of topological graph theory. Enumeration formulas were developed for a large class of graph covering projections. The purpose of this paper is to count graph covering projections of graphs such that the corresponding covering space is a connected graph. The main tool of the enumeration is Polya's theorem. Key words. graph covering, enumeration, Polya's theorem AMS subject classifications. 05C10, 05C30, 57M10 PII. S0895480195293873 1. Introduction. In this paper, we consider simple undirected graphs. As usual, the vertex set and the edge set of the graph G are denoted by V (G) and E(G), respectively. An r-to-one graph epimorphism p : H # G which sends the neighbors of each vertex x # V (H) bijectively to the neighbors of p(x) # V (G) is called an r-fold covering projection of G. The graph H is the covering graph, and the graph G is the base graph of p. Topologically speaking, p is a local hom...

By applying the Tutte decomposition of 2--connected graphs into 3--block trees we provide unique structural characterizations of several classes of 2--connected graphs, including minimally 2--connected graphs, minimally 2--edge--connected graphs, critically 2--connected graphs, critically 2--edge--connected graphs, 3--edge--connected graphs, 2--connected cubic graphs and 3--connected cubic graphs. We also give a characterization of minimally 3connected graphs. 1

N. G. de Bruijn and J. W. Nienhuys. Preface These are Nienhuys ’ lecture notes on a course given by de Bruijn in the 1980s.

NOVEMBER 2003This thesis is dedicated to the memory of a wonderful woman, my mother, Vicki Munn. Acknowledgements I offer the following people and organizations heartfelt appreciation for their contributions to this work and their support during the period of my thesis. The two research teams which I had the pleasure of being part: LaCIM (Université du Québec à Montréal) and Projet Algorithmes (Inria, Rocquencourt, France). Both are thriving incubators of combinatorics with warm, human aspects, and regular coffee. I thank also the students and postdocs of both groups, such as Marianne Durand, Cédric Lamathe, and Ludovic Meunier, for their humour and camaraderie; A special separate mention must be made for les assistantes extraordinaires Lise

The automorphisms of a graph act naturally on its set of labeled imbeddings to produce its unlabeled imbeddings. The imbedding sum of a graph is a polynomial that contains useful information about a graph’s labeled and unlabeled imbeddings. In particular, the polynomial enumerates the number of different ways the unlabeled imbeddings can be vertex colored and enumerates the labeled and unlabeled imbeddings by their symmetries.