Commutative association schemes

Association schemes were originally introduced by Bose and his co-workers in the design of statistical experiments. Since that point of inception, the concept has proved useful in the study of group actions, in algebraic graph theory, in algebraic coding theory, and in areas as far afield as knot theory and numerical integration. This branch of the theory, viewed in this collection of surveys as the "commutative case," has seen significant activity in the last few decades. The goal of the present survey is to discuss the most important new developments in several directions, including Gelfand pairs, cometric association schemes, Delsarte Theory, spin models and the semidefinite programming technique. The narrative follows a thread through this list of topics, this being the contrast between combinatorial symmetry and group-theoretic symmetry, culminating in Schrijver's SDP bound for binary codes (based on group actions) and its connection to the Terwilliger algebra (based on combinatorial symmetry). We propose this new role of the Terwilliger algebra in Delsarte Theory as a central topic for future work.Comment: 36 page

Applications of the Combinatorial Nullstellensatz on bipartite graphs.

The Combinatorial Nullstellensatz can be used to solve certain problems in combinatorics. However, one of the major complications in using the Combinatorial Nullstellensatz is ensuring that there exists a nonzero monomial. This dissertation looks at applying the Combinatorial Nullstellensatz to finding perfect matchings in bipartite graphs. The first two chapters provide background material covering topics such as linear algebra, group theory, graph theory and even the discrete Fourier transform. New results start in the third chapter, showing that the Combinatorial Nullstellensatz can be used to solve the problem of finding perfect matchings in bipartite graphs. Using the Combinatorial Nullstellensatz also allows for a vice use of matroid intersection to find the nonzero monomial. By also applying the uncertainty principle, the number of perfect matchings in a bipartite graph can be bound. The fourth chapter examines properties of the polynomials created in the use of the Combinatorial Nullstellensatz to find perfect matchings in bipartite graphs. Many of the properties of the polynomials have analogous properties for the transforms of the polynomials, which are also examined. These properties often relate back to the structure of the graph which gave rise to the polynomial. The fifth chapter provides an application of the results. Since finding a nonzero monomial can be difficult and the polynomials created in this dissertation give polynomials with such a nonzero monomial the application shows how certain polynomials can be rewritten in terms of the matching polynomials. Such a rewriting may permit an easy method to find a nonzero monomial so that the Combinatorial Nullstellensatz can be applied to the polynomial. Finally, the fifth chapter concludes with some open problems that may be areas of further research

Computational complexity of reconstruction and isomorphism testing for designs and line graphs

Graphs with high symmetry or regularity are the main source for experimentally hard instances of the notoriously difficult graph isomorphism problem. In this paper, we study the computational complexity of isomorphism testing for line graphs of $t$-$(v,k,\lambda)$ designs. For this class of highly regular graphs, we obtain a worst-case running time of $O(v^{\log v + O(1)})$ for bounded parameters $t,k,\lambda$. In a first step, our approach makes use of the Babai--Luks algorithm to compute canonical forms of $t$-designs. In a second step, we show that $t$-designs can be reconstructed from their line graphs in polynomial-time. The first is algebraic in nature, the second purely combinatorial. For both, profound structural knowledge in design theory is required. Our results extend earlier complexity results about isomorphism testing of graphs generated from Steiner triple systems and block designs.Comment: 12 pages; to appear in: "Journal of Combinatorial Theory, Series A