Let Γ denote a distance-regular graph with classical parameters (D, b, α, β) and D ≥ 3. Assume the intersection numbers a1 = 0 and a2 6= 0. We show Γ is 3-bounded in the sense of the article [D-bounded distance-regular graphs, European Journal of Combinatorics(1997)18, 211-229].

Under the assumption that δ is a Woodin cardinal and GCH holds, I show that if F is any class function from the regular cardinals to the cardinals such that (1) κ < cf(F (κ)), (2) κ < λ implies F (κ) ≤ F (λ), and (3) δ is closed under F , then there is a cofinality-preserving forcing extension in which 2 = F (γ) for each regular cardinal γ < δ, and in which δ remains Woodin. Unlike the analogou...

Let Γ be a triangle-free distance-regular graph with diameter d ≥ 3, valency k ≥ 3 and intersection number a2 6= 0. Assume Γ has an eigenvalue with multiplicity k. We show that Γ is 1-homogeneous in the sense of Nomura when d = 3 or when d ≥ 4 and a4 = 0. In the latter case we prove that Γ is an antipodal cover of a strongly regular graph, which means that it has diameter 4 or 5. For d = 5 the ...

Let Γ denote a distance-regular graph with classical parameters (D, b, α, β) and D ≥ 3. Assume the intersection numbers a1 = 0 and a2 6= 0. We show Γ is 3-bounded in the sense of the article [D-bounded distance-regular graphs, European Journal of Combinatorics(1997)18, 211-229].

The weak regular coherence is a coarse property of a finitely generated group Γ. It was introduced by G. Carlsson and this author to play the role of a weakening of Waldhausen’s regular coherence as part of computation of the integral K-theoretic assembly map. A new class of metric spaces (sFDC) was introduced recently by A. Dranishnikov and M. Zarichnyi. This class includes most notably the sp...

A strongly regular graph with λ = 0 and μ = 3 is of degree 3 or 21. The automorphisms of prime order and the subgraphs of their fixed points are described for a strongly regular graph Γ with parameters (162, 21, 0, 3). In particular, the inequality |G/O(G)| ≤ 2 holds true for G = Aut(Γ).

Let T be a k-regular tree (k ≥ 3) and A = Aut(T ) its automorphism group. We analyze a generic finitely generated subgroup Γ of A. We show that Γ is free and establish a trichotomy on the closure Γ of Γ in A. It turns out that Γ is either discrete, compact or has index at most 2 in A.

An indeterminate string x = x[1..n] on an alphabet Σ is a sequence of nonempty subsets of Σ; x is said to be regular if every subset is of size one. A proper substring u of regular x is said to be a cover of x iff for every i ∈ 1..n, an occurrence of u in x includes x[i]. The cover array γ = γ[1..n] of x is an integer array such that γ[i] is the longest cover of x[1..i]. Fifteen years ago a com...

Let T be a k-regular tree (k ≥ 3) and A = Aut(T) its automorphism group. We analyze a generic finitely generated subgroup Γ of A. We show that Γ is free and establish a trichotomy on the closure Γ of Γ in A. It turns out that Γ is either discrete, compact or has index at most 2 in A.