Jack H. Koolen

1-Homogeneous Graphs with Cocktail Party μ-Graphs

Let Γ be a graph with diameter d ≥ 2. Recall Γ is 1-homogeneous (in the sense of Nomura) whenever for every edge xy of Γ the distance partition{{z ∈ V(Γ) | ∂(z, y) = i, ∂(x, z) = j} | 0 ≤ i, j ≤ d}is equitable and its parameters do not depend on the edge xy. Let Γ be 1-homogeneous. Then Γ is distance-regular and also locally strongly regular with parameters (v′,k′,λ′,μ′), where v′ = k, k′ = a1, (v′ − k′ − 1)μ′ = k′(k′ − 1 − λ′) and c2 ≥ μ′ + 1, since a μ-graph is a regular graph with valency μ′. If c2 = μ′ + 1 and c2 ≠ 1, then Γ is a Terwilliger graph, i.e., all the μ-graphs of Γ are complete. In [11] we classified the Terwilliger 1-homogeneous graphs with c2 ≥ 2 and obtained that there are only three such examples. In this article we consider the case c2 = μ′ + 2 ≥ 3, i.e., the case when the μ-graphs of Γ are the Cocktail Party graphs, and obtain that either λ′ = 0, μ′ = 2 or Γ is one of the following graphs: (i) a Johnson graph J(2m, m) with m ≥ 2, (ii) a folded Johnson graph J¯(4m, 2m) with m ≥ 3, (iii) a halved m-cube with m ≥ 4, (iv) a folded halved (2m)-cube with m ≥ 5, (v) a Cocktail Party graph Km × 2 with m ≥ 3, (vi) the Schläfli graph, (vii) the Gosset graph.

Nonexistence of some Antipodal Distance-regular Graphs of Diameter Four

We find an inequality involving the eigenvalues of a regular graph; equality holds if and only if the graph is strongly regular. We apply this inequality to the first subconstituents of a distance-regular graph and obtain a simple proof of the fundamental bound for distance-regular graphs, discovered by Juri?i?, Koolen and Terwilliger. Using this we show that for distance-regular graphs with certain intersection arrays, the first subconstituent graphs are strongly regular. From these results we prove the nonexistence of distance-regular graphs associated to 20 feasible intersection arrays from the book Distance-Regular Graphs by Brouwer, Cohen and Neumaier .

The coherency index

Abstract A coherent decomposition of a metric is, in general, a natural decomposition of the metric into a sum of simpler metrics. A metric that has only a trivial coherent decomposition is called prime. In this paper we give a formula for an index, called the coherency index, which allows us to prove that there exist only finitely many prime metrics (up to multiplication by a positive scalar) on a finite set. Moreover, the formula that we give for the coherency index also provides us with a computational tool by which one can compute coherent decompositions of metrics into primes.

A generalization of an inequality of Brouwer-Wilbrink

Brouwer and Wilbrink showed that t + 1 ≤ (s2 + 1)cd-1 holds for a regular near 2d-gon of order (s, t) with s ≥ 2 and where the diameter d is even.In this note we generalize their inequality to all diameter.

An improvement of the Ivanov bound

AbstractLet Γ be a distance-regular graph of diameterd, valencyk andr=max{i|(ci,bi)=(c1,b1)}. In this paper, we prove that

A bound for the number of columns ℓ( c,a,b ) in the intersection array of a distance-regular graph

d< \frac{1}{2}k^3 r.

A New Distance-Regular Graph Associated to the Mathieu Group M {10}

Abstract.Let Γ be a regular near polygon of order (s,t) with s>1 and t≥3. Let d be the diameter of Γ, and let r:= max{i∣(ci,ai,bi)=(c1,a1,b1)}. In this note we prove several inequalities for Γ. In particular, we show that s is bounded from above by function in t if We also consider regular near polygons of order (s,3).