E.R. van Dam

On almost distance-regular graphs

Distance-regular graphs are a key concept in Algebraic Combinatorics and have given rise to several generalizations, such as association schemes. Motivated by spectral and other algebraic characterizations of distance-regular graphs, we study almost distance-regular graphs. We use this name informally for graphs that share some regularity properties that are related to distance in the graph. For example, a known characterization of a distance-regular graph is the invariance of the number of walks of given length between vertices at a given distance, while a graph is called walk-regular if the number of closed walks of given length rooted at any given vertex is a constant. One of the concepts studied here is a generalization of both distance-regularity and walk-regularity called m-walk-regularity. Another studied concept is that of m-partial distance-regularity or, informally, distance-regularity up to distance m. Using eigenvalues of graphs and the predistance polynomials, we discuss and relate these and other concepts of almost distance-regularity, such as their common generalization of (@?,m)-walk-regularity. We introduce the concepts of punctual distance-regularity and punctual walk-regularity as a fundament upon which almost distance-regular graphs are built. We provide examples that are mostly taken from the Foster census, a collection of symmetric cubic graphs. Two problems are posed that are related to the question of when almost distance-regular becomes whole distance-regular. We also give several characterizations of punctually distance-regular graphs that are generalizations of the spectral excess theorem.

Characterizing distance-regularity of graphs by the spectrum

We characterize the distance-regular Ivanov-lvanov-Faradjev graph from the spectrum, and construct cospectral graphs of the Johnson graphs, Doubled Odd graphs, Grassmann graphs, Doubled Grassmann graphs, antipodal covers of complete bipartite graphs, and many of the Taylor graphs. We survey the known results on cospectral graphs of the Hamming graphs, and of all distance-regular graphs on at most 70 vertices.

Some implications on amorphic association schemes

We give an overview of results on amorphic association schemes. We give the known constructions of such association schemes, and enumerate most such association schemes on up to 49 vertices. Special attention is paid to cyclotomic association schemes. We give several results on when a strongly regular decomposition of the complete graph is an amorphic association scheme. This includes a new proof of the result that a decomposition of the complete graph into three strongly regular graphs is an amorphic association scheme, and the new result that a strongly regular decomposition of the complete graph for which the union of any two relations is again strongly regular must be an amorphic association scheme.

Combinatorial designs with two singular values I: uniform multiplicative designs

In this and a sequel paper (Combinatorial designs with two singular values. II. Partial geometric designs, preprint) we study combinatorial designs whose incidence matrix has two distinct singular values. These generalize 2-(v, k, λ) designs, and include partial geometric designs and uniform multiplicative designs. Here we study the latter, which are precisely the nonsingular designs. We classify all such designs with smallest singular value at most √2, generalize the Bruck-Ryser-Chowla conditions, and enumerate, partly by computer, all uniform multiplicative designs on at most 30 points.

Association Schemes Related to Kasami Codes and KerdockSets

Two new infinite series of imprimitive 5-class association schemes are constructed. The first series of schemes arises from forming, in a special manner, two edge-disjoint copies of the coset graph of a binary Kasami code (double error-correcting BCH code). The second series of schemes is formally dual to the first. The construction applies vector space duality to obtain a fission scheme of a subscheme of the Cameron-Seidel 3-class scheme of linked symmetric designs derived from Kerdock sets and quadratic forms over GF(2).

A Nonregular Analogue of Conference Graphs

We prove some results on graphs with three eigenvalues, not all integral; these are a natural generalization of the strongly regular conference graphs. We derive a Bruck?Ryser type condition and construct some (nonregular) examples.

Uniformly Packed Codes and More Distance Regular Graphs from Crooked Functions

AbstractLet V and W be n-dimensional vector spaces over GF(2). A function Q : V → W is called crooked (a notion introduced by Bending and Fon-Der-Flaass) if it satisfies the following three properties:n

Fissions of Classical Self-Dual Association Schemes

begin{gathered} Q(0) = 0; hfill Q(x) + Q(y) + Q(z) + Q(x + y + z) ne 0{text{ for any three distinct }}x,y,z; hfill Q(x) + Q(y) + Q(z) + Q(x + a) + Q(y + a) + Q(z + a) ne 0{text{ if }}a ne 0{text{ }}(x,y,z{text{ arbitrary}}). hfill end{gathered}