Properties of Commutative Association Schemes derived by FGLM Techniques

Association schemes are combinatorial objects that allow us solve problems in several branches of mathematics. They have been used in the study of permutation groups and graphs and also in the design of experiments, coding theory, partition designs etc. In this paper we show some techniques for computing properties of association schemes. The main framework arises from the fact that we can characterize completely the Bose-Mesner algebra in terms of a zero-dimensional ideal. A Gr\"obner basis of this ideal can be easily derived without the use of Buchberger algorithm in an efficient way. From this statement, some nice relations arise between the treatment of zero-dimensional ideals by reordering techniques (FGLM techniques) and some properties of the schemes such as P-polynomiality, and minimal generators of the algebra.Comment: 12 pages, to appear in the International Journal of Algebra and Computatio

Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms. II. Program Implementation of the Weisfeiler-Leman Algorithm

The stabilization algorithm of Weisfeiler and Leman has as an input any square matrix A of order n and returns the minimal cellular (coherent) algebra W(A) which includes A. In case when A=A(G) is the adjacency matrix of a graph G the algorithm examines all configurations in G having three vertices and, according to this information, partitions vertices and ordered pairs of vertices into equivalence classes. The resulting construction allows to associate to each graph G a matrix algebra W(G):= W(A(G))$ which is an invariant of the graph G. For many classes of graphs, in particular for most of the molecular graphs, the algebra W(G) coincides with the centralizer algebra of the automorphism group aut(G). In such a case the partition returned by the stabilization algorithm is equal to the partition into orbits of aut(G). We give algebraic and combinatorial descriptions of the Weisfeiler--Leman algorithm and present an efficient computer implementation of the algorithm written in C. The results obtained by testing the program on a considerable number of examples of graphs, in particular on some chemical molecular graphs, are also included.Comment: Arxiv version of a preprint published in 199

Constructive and analytic enumeration of circulant graphs with p3p^3 vertices; p=3,5p=3,5

Two methods, structural (constructive) and multiplier (analytical), of exact enumeration of undirected and directed circulant graphs of orders 27 and 125 are elaborated and represented in detail here together with intermediate and final numerical data. The first method is based on the known useful classification of circulant graphs in terms of $S$-rings and results in exhaustive listing (with the use of COCO and GAP) of all corresponding $S$-rings of the indicated orders. The latter method is conducted in the framework of a general approach developed earlier for counting circulant graphs of prime-power orders. It is a Redfield--P\'olya type of enumeration based on an isomorphism criterion for circulant graphs of such orders. In particular, five intermediate enumeration subproblems arise, which are refined further into eleven subproblems of this type (5 and 11 are, not accidentally, the 3d Catalan and 3d little Schr\"oder numbers, resp.). All of them are resolved for the four cases under consideration (again with the use of GAP). We give a brief survey of some background theory of the results which form the basis of our computational approach. Except for the case of undirected circulant graphs of orders 27, the numerical results obtained here are new. In particular the number (up to isomorphism) of directed circulant graphs of orders 27, regardless of valency, is shown to be equal to 3,728,891 while 457 of these are self-complementary. Some curious and rather unexpected identities are established between intermediate valency-specified enumerators (both for undirected and directed circulant graphs) and their validity is conjectured for arbitrary cubed odd prime $p^3$. We believe that this research can serve as the crucial step towards explicit uniform enumeration formulae for circulant graphs of orders $p^3$ for arbitrary prime $p>2$.Comment: 72 pages, 5 figure, 12 tables, 46 references, 2 appendice

Automorphism groups of rational circulant graphs through the use of Schur rings

The paper concerns the automorphism groups of Cayley graphs over cyclic groups which have a rational spectrum (rational circulant graphs for short). With the aid of the techniques of Schur rings it is shown that the problem is equivalent to consider the automorphism groups of orthogonal group block structures of cyclic groups. Using this observation, the required groups are expressed in terms of generalized wreath products of symmetric groups

On the Enumeration of Circulant Graphs of Prime-Power Order: the case of p3p^3

A well-known problem in Algebraic Combinatorics, is the enumeration of circulant graphs. The failure of Adam's Conjecture for such graphs with order containing a repeated prime, led researchers to investigate the problem using two different methods, namely the multiplier method and the structural method. The former makes use of isomorphism theorems whereas the latter involves Schur rings. Both these methods have already been used to count the number of non-isomorphic circulants of order $p^2$. This research focuses on the extension of these two methods to enumerate circulants of order $p^3$, in particular for $p=3$ and $p=5$, through the use of the computer package GAP.Comment: 160 pages; 8 tables; 11 figures; two appendice