Akira Hiraki

Distance-regular subgraphs in a distance-regular graph, II

Let ? be a distance-regular graph without induced subgraphsK2,1,1andr=max {j? (cj,aj,bj)= (c1,a1,b1)}. We give a necessary and sufficient condition for the existence of a strongly closed subgraph which is (cr+1+ar+1)-regular of diameterr+1 containing a given pair of vertices at distancer+1.

A Distance-Regular Graph with Strongly Closed Subgraphs

AbstractLet Γ be a distance-regular graph of diameter d, valency k and r := maxi | (ci,bi) = (c1,b1). Let q be an integer with r + 1 ≤ q ≤ d − 1.In this paper we prove the following results: Theorem 1Suppose for any pair of vertices at distance q there exists a strongly closed subgraph of diameter q containing them. Then for any integer i with 1 ≤ i ≤ qand for any pair of vertices at distance i there exists a strongly closed subgraph of diameter i containing them.Theorem 2If r ≥ 2, thenc2r+3 ≠ 1.As a corollary of Theorem 2 we have d ≤ k2(r + 1) if r ≥ 2.

Strongly Closed Subgraphs in a Regular Thick Near Polygon

In this paper we show that a regular thick near polygon has a tower of regular thick near sub-polygons as strongly closed subgraphs if the diameter d is greater than the numerical girth g.

An improvement of the Boshier-Nomura bound

Abstract We show that the number of columns ( c i , a i , b i ) = (1, 1, k − 2) in the intersection arrays of distance-regular graphs is at most three if the column (1, 0, k − 1) exists. This improves the Bosheir-Nomura bound from four to three.

Delsarte clique graphs

In this paper, we consider the class of Delsarte clique graphs, i.e. the class of distance-regular graphs with the property that each edge lies in a constant number of Delsarte cliques. There are many examples of Delsarte clique graphs such as the Hamming graphs, the Johnson graphs and the Grassmann graphs. Our main result is that, under mild conditions, for given s>=2 there are finitely many Delsarte clique graphs which contain Delsarte cliques with size s+1. Further we classify the Delsarte clique graphs with small s.

Distance-Regular Graphs of Valency 6 and a1 = 1

We give a complete classification of distance-regular graphs of valency 6 and a1 = 1.

A characterization of the doubled Grassmann graphs, the doubled odd graphs, and the odd graphs by strongly closed subgraphs

We characterize the doubled Grassmann graphs, the doubled Odd graphs, and the Odd graphs by the existence of sequences of strongly closed subgraphs.

A Circuit Chasing Technique in a Distance-regular Graph with Triangles

We give an example of circuit chasing on a distance-regular graph which has triangle. In particular, we show that the number of columns (ci, ai, bi) = (2, 2a, e) in the intersection array of a distance-regular graph is at most 1, if all the preceding columns are (1, a, b).

Improving diameter bounds for distance-regular graphs

In this paper we study the sequence (ci)0 ≤ i ≤ d for a distance-regular graph. In particular we show that if d ≥ 2j and cj > 1 then c2j-1 > cj holds. Using this we give improvements on diameter bounds by A. Hiraki, J.H. Koolen [An improvement of the Ivanov bound, Ann. Comb. 2 (2) (1998) 131-135], and L. Pyber [A bound for the diameter of distance-regular graphs, Combinatorica 19 (4) (1999) 549-553], respectively, by applying this inequality.

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.