This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.

We give a new upper bound on the maximum size A q (n, d) of a code of word length n and minimum Hamming distance at least d over the alphabet of q 3 letters. By block-diagonalizing the Terwilliger algebra of the nonbinary Hamming scheme, the bound can be calculated in time polynomial in n using semidefinite programming. For q = 3, 4, 5 this gives several improved upper bounds for concrete values of n and d. This work builds upon previous results of Schrijver [A. Schrijver, New code upper bounds from the Terwilliger algebra and semidefinite programming, IEEE Trans. Inform. Theory 51 (2005) 2859-2866] on the Terwilliger algebra of the binary Hamming scheme.

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

Brouwer, Godsil, Koolen and Martin [Width and dual width of subsets in polynomial association schemes, J. Combin. Theory Ser. A 102 (2003) 255-271] introduced the width w and the dual width w * of a subset in a distance-regular graph and in a cometric association scheme, respectively, and then derived lower bounds on these new parameters. For instance, subsets with the property w + w * = d in a cometric distance-regular graph with diameter d attain these bounds. In this paper, we classify subsets with this property in Grassmann graphs, bilinear forms graphs and dual polar graphs. We use this information to establish the Erdős-Ko-Rado theorem in full generality for the first two families of graphs.

Abstract. In this paper, we obtain certain discrete orthogonal polynomials expressed in terms of the (d + 1, 2(d + 1))-hypergeometric functions, from the eigenmatrices of character algebras.

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The Mean King's problem with mutually unbiased bases is reconsidered for arbitrary d-level systems. Hayashi, Horibe and Hashimoto [Phys. Rev. A 71, 052331 (2005)] related the problem to the existence of a maximal set of d − 1 mutually orthogonal Latin squares, in their restricted setting that allows only measurements of projection-valued measures. However, we then cannot find a solution to the problem when e.g., d = 6 or d = 10. In contrast to their result, we show that the King's problem always has a solution for arbitrary levels if we also allow positive operator-valued measures. In constructing the solution, we use orthogonal arrays in combinatorial design theory.

a b s t r a c t We consider finite analogues of Euclidean graphs in a more general setting than that considered in [A. Medrano, P. Myers, H.M. Stark, A. Terras, Finite analogues of Euclidean space, J. Comput. Appl. Math. 68 (1996) 221-238] and we obtain many new examples of Ramanujan graphs. In order to prove these results, we use the previous work of [W.M. Kwok, Character tables of association schemes of affine type, European J. Combin. 13 (1992) 167-185] calculating the character tables of certain association schemes of affine type.A key observation is that we can obtain better estimates for the ordinary Kloosterman sum K (a, b; q). In particular, we always achieve |K (a, b; q)| < 2 √ q, and |K (a, b; q)| ≤ 2 √ q − 2 in many (but not all) of the cases, instead of the well known |K (a, b; q)| ≤ 2 √ q. Also, we use the ideas of controlling association schemes, and the Ennola type dualities, in our previous work on the character tables of commutative association schemes. The method in this paper will be used to construct many more new examples of families of Ramanujan graphs in the subsequent paper.