D-BOUNDED DISTANCE-REGULAR GRAPHS

AbstractLet Γ=(X, R) denote a distance-regular graph with diameterD≥3 and distance functionδ. A (vertex) subgraph Δ⊆Xis said to beweak-geodetically closedwhenever for allx, y∈Δ and allz∈X,δ(x, z)+δ(z, y)≤δ(x, y)+1→z∈Δ.Γis said to beD-boundedwhenever, for allx, y ∈X, xandyare contained in a common regular weak-geodetically closed subgraph of diameterδ(x, y).Assume that Γ isD-bounded. LetP(Γ)denote the poset the elements of which are the weak-geodetically closed subgraphs of Γ with partial order by reverse inclusion. We obtain new inequalities for the intersection numbers of Γ; equality is obtained in each of these inequalities iff the intervals inP(Γ)are modular. Moreover, we show this occurs if Γ has classical parameters andD≥4.We obtain the following corollary without assuming Γ to beD-bounded:COROLLARY.Let Γ denote a distance-regular graph with classical parameters (D, b, α, β) and D≥4. Suppose that b<-1, and suppose the intersection numbers a1≠0 and c2>1. Thenβ=α1+bD1−

3-bounded property in a triangle-free distance-regular graph

Let $\Gamma$ denote a distance-regular graph with classical parameters $(D, b, \alpha, \beta)$ and $D\geq 3$. Assume the intersection numbers $a_1=0$ and $a_2\not=0$. We show $\Gamma$ is 3-bounded in the sense of the article [D-bounded distance-regular graphs, European Journal of Combinatorics(1997)18, 211-229].Comment: 13 page

Lattices generated by join of strongly closed subgraphs in d-bounded distance-regular graphs

AbstractLet Γ be a d-bounded distance-regular graph with diameter d⩾3. Suppose that P(x) is a set of all strongly closed subgraphs containing x and that P(x,i) is a subset of P(x) consisting of all elements of P(x) with diameter i. Let L′(x,i) be the set generated by all joins of the elements in P(x,i). By ordering L′(x,i) by inclusion or reverse inclusion, L′(x,i) is denoted by LO′(x,i) or LR′(x,i). We prove that LO′(x,i) and LR′(x,i) are both finite atomic lattices, and give the conditions for them both being geometric lattices. We also give the eigenpolynomial of LO′(x,i)

Pooling spaces associated with finite geometry

AbstractMotivated by the works of Ngo and Du [H. Ngo, D. Du, A survey on combinatorial group testing algorithms with applications to DNA library screening, DIMACS Series in Discrete Mathematics and Theoretical Computer Science 55 (2000) 171–182], the notion of pooling spaces was introduced [T. Huang, C. Weng, Pooling spaces and non-adaptive pooling designs, Discrete Mathematics 282 (2004) 163–169] for a systematic way of constructing pooling designs; note that geometric lattices are among pooling spaces. This paper attempts to draw possible connections from finite geometry and distance regular graphs to pooling spaces: including the projective spaces, the affine spaces, the attenuated spaces, and a few families of geometric lattices associated with the orbits of subspaces under finite classical groups, and associated with d-bounded distance-regular graphs

Distance-regular graphs

This is a survey of distance-regular graphs. We present an introduction to distance-regular graphs for the reader who is unfamiliar with the subject, and then give an overview of some developments in the area of distance-regular graphs since the monograph 'BCN' [Brouwer, A.E., Cohen, A.M., Neumaier, A., Distance-Regular Graphs, Springer-Verlag, Berlin, 1989] was written.Comment: 156 page

A characterization of the Hamming graph by strongly closed subgraphs

AbstractThe Hamming graph H(d,q) satisfies the following conditions: (i)For any pair (u,v) of vertices there exists a strongly closed subgraph containing them whose diameter is the distance between u and v. In particular, any strongly closed subgraph is distance-regular.(ii)For any pair (x,y) of vertices at distance d−1 the subgraph induced by the neighbors of y at distance d from x is a clique of size a1+1.In this paper we prove that a distance-regular graph which satisfies these conditions is a Hamming graph