Spreads of finite symplectic, orthogonal and unitary vector spaces are used to construct new strongly regular graphs having the same parameters as the perpendicularity graphs of the underlying vector spaces. Some of the graphs are. related to partial geometries, while others produce interesting symmetric designs.

We construct distance-regular graphs with the same classical parameters as the Grassmann graphs on the e-dimensional subspaces of a (2e+1)-dimensional space over an arbitrary finite field. This provides the first known family of non-transitive distance-regular graphs with unbounded diameter.

In this paper we discuss a relationship between the following two algebras: (i) the subconstituent algebra T of a distance-regular graph that has q-Racah type; (ii) the q-tetrahedron algebra ⊠q which is a q-deformation of the three-point sl2 loop algebra. Assuming that every irreducible T -module is thin, we display an algebra homomorphism from ⊠q into T and show that T is generated by the imag...

In 1971, Nash-Williams proved that if G is a simple 2-connected graph on n vertices having minimum degree at least 3 (n+ 2), then any longest cycle C in G is also edge-domininating; that is, each edge of G has at least one end-vertex incident with a vertex of C. We say that a circuit C in a matroid M is dominating if each component of M/C has rank at most one. In this paper, we show that an ana...

We show that the contact process on a random d-regular graph initiated by a single infected vertex obeys the “cutoff phenomenon” in its supercritical phase. In particular, we prove that, when the infection rate is larger than the lower critical value of the contact process on the infinite d-regular tree, there are positive constants C, p depending on the infection rate such that for any ε > 0, ...

The goal of this paper is to establish a connection between two classical models of random graphs: the random graph G(n, p) and the random regular graph Gd(n). This connection appears to be very useful in deriving properties of one model from the other. In particular, one immediately obtains one-line proofs of several highly non-trivial and recent results on Gd(n).

In this paper we construct transitive t-designs from the linear groups L(2, q), q ≤ 23. Thereby we classify t-designs, t ≥ 2, admitting a transitive action of the linear groups L(2, q), q ≤ 23, up to 35 points and obtained numerous transitive designs, for 36 ≤ v ≤ 55. In many cases we proved the existence of t-designs with certain parameter sets. Among others we constructed t-designs with param...

In this paper, we introduce distance-regular graphs and develop the intersection algebra for these graphs which is based upon its intersection numbers. We discuss results following from the definition of the intersection algebra. We investigate two examples of distance-regular graphs and show how these results apply. Finally, we introduce parameters that determine intersection numbers. We inves...

The chromatic polynomials of ‘bracelets’ can be studied by means of a theory based on representations of the symmetric group. This paper contains a detailed study of the theory as it relates to one type of bracelet. The underlying theory is presented rather more clearly than hitherto, and some surprising features are exhibited. The methods involve an extension of the standard theory of distance...

Combining results on quadrics in projective geometries with an algebraic interplay between finite fields and Galois rings, we construct the first known family of partial difference sets with negative Latin square type parameters in nonelementary abelian groups, the groups Z 4 × Z4`−4k 2 for all k when ` is odd and for all k < ` when ` is even. Similarly, we construct partial difference sets wit...