Sylvia A. Hobart

The Structure of Nonthin Irreducible T-modulesof Endpoint 1: Ladder Bases and Classical Parameters

Building on the work of Terwilliger, we find the structure of nonthin irreducible T-modules of endpoint 1 for P- and Q-polynomial association schemes with classical parameters. The isomorphism class of such a given module is determined by the intersection numbers of the scheme and one additional parameter which must be an eigenvalue for the first subconstituent graph. We show that these modules always have what we call a ladder basis, and find the structure explicitly for the bilinear, Hermitean, and alternating forms schemes.

A Note on A Family of Directed Strongly Regular Graphs

where J is the all ones matrix and I is the identity. These matrix conditions are equivalent to the combinatorial conditions that the graph is both inand out-regular, and that the number of directed 2-paths from a vertex x to a vertex y is t if x = y, λ if x → y, and μ otherwise. Recently, these graphs were studied by Klin et al. [2], including some new constructions and a list of feasible parameters. In this note, we will use Cayley graphs to construct a new infinite family of dsrg’s. For any subset S of a group G, the (directed) Cayley graph of G with respect to S, denoted C(G, S), is the directed graph with vertex set G where g→ h if and only if hg−1 ∈ S. We can determine whether a Cayley graph is a dsrg using the group ring ZG. For any subset T ⊆ G, define T ∈ ZG by T = ∑ t∈T t . LEMMA 1. The Cayley graph C(G, S), is a (v, k, μ, λ, t)-dsrg if and only if the equation S 2 = te + λS + μ(G − e − S) holds in ZG.

Extended Partial Geometries: Nets and Dual Nets

Extended partial geometries ( EpGs ) are natural generalisations of extended generalised quadrangles. We develop some general results for any EpG , and then consider the special cases of nets and dual nets in detail. Extended dual nets have diameter at most 2, and are almost characterised, with examples, but extended nets are both more complex and perhaps more interesting: there are many examples, but also many open cases, and ‘reasonable’ conjectures.

Representations of directed strongly regular graphs

We develop a theory of representations in R^m for directed strongly regular graphs, which gives a new proof of a nonexistence condition of Jorgensen [L.K. Jorgensen, Non-existence of directed strongly regular graphs, Discrete Math. 264 (2003) 111-126]. We also describe some new constructions.

Near subgroups of groups

Abstract A finite subset A of a group is a near subgroup if the number of ordered pairs ( x , y ) ϵ A 2 with xy ⊂ A is at most / vb A / vb . We show here that if / vb A / vb ⩾ 5, then A is a near subgroup if and only if A ∪ {g} is a subgroup for some group element g . We also classify the counterexamples if / vb A / vb ⩽ 4.

On designs related to coherent configurations of type 2 2 4

Abstract Coherent configurations of type ( 2 2 4 ) correspond to certain complementary pairs of 2-designs with two or three intersection sizes, which are investigated in this paper. We establish relations on the parameters of such designs, and use them to show the Witt design L (5, 8, 24) is determined by the association scheme on its blocks, and to characterize the family of designs based on systems of linked symmetric designs.

Reconstructing a Generalized Quadrangle from its Distance Two Association Scheme

Payne [4] constructed an association scheme from a generalized quadrangle with a quasiregular point. We show that an association scheme with appropriate parameters and satisfying an assumption about maximal cliques must be one of these schemes arising from a generalized quadrangle.

Krein conditions for coherent configurations

Abstract A coherent configuration can be thought of as a set of association schemes linked by additional relations. We develop a generalization of the Krein conditions which applies to coherent configurations and which takes into account the additional information contained in the relations between the schemes. We then apply it to quasisymmetric designs and strongly regular designs to obtain inequalities on the parameters. We also give examples to show that these conditions are stronger than the usual Krein conditions on the association schemes contained in the configuration.

A characterization of t-designs in terms of the inner distribution

We derive an inequality for 0-designs which holds with equality iff the design is a t -design. This inequality is then applied to the case of regular designs; that is, designs in which the number of blocks intersecting a given block in a given number of points is a constant depending only on the number of points. This analysis is used to characterize the Steiner system S (4, 7, 23) in terms of the derived design at two points.