An equitable partition for a distance-regular graph of negative type

An equitable partition for a distance-regular graph of negative type

Classical Distance-Regular Graphs of Negative Type

Brouwer, Cohen, and Neumaier found that the intersection numbers of most known families of distance-regular graphs could be described in terms of four parameters (d, b, :, ;) [2, pp. ix, 193]. They invented the term classical to describe those graphs for which this could be done. All classical distance-regular graphs with b=1 are classified by Y. Egawa, A. Neumaier, and P. Terwilliger in a sequ...

The Terwilliger Algebra of a Distance-regular Graph of Negative Type

Let Γ denote a distance-regular graph with diameter D ≥ 3. Assume Γ has classical parameters (D, b, α, β) with b < −1. Let X denote the vertex set of Γ and let A ∈ MatX(C) denote the adjacency matrix of Γ. Fix x ∈ X and let A ∈ MatX(C) denote the corresponding dual adjacency matrix. Let T denote the subalgebra of MatX(C) generated by A,A . We call T the Terwilliger algebra of Γ with respect to ...

Subgraphs Graph in a Distance-regular Graph

Let Γ denote a D-bounded distance-regular graph, where D ≥ 3 is the diameter of Γ. For 0 ≤ s ≤ D − 3 and a weak-geodetically closed subgraph ∆ of Γ with diameter s, define a graph G(∆) whose vertex set is the collection of all weak-geodetically closed subgraphs of diameter s+2 containing ∆, and vertex Ω is adjacent to vertex Ω′ in G if and only if Ω∩Ω′ as a subgraph of Γ has diameter s+1. We sh...

A Questionable Distance-Regular Graph

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...