If n ⩾ 2d ⩾ 6 and q ⩾ 4, the distance-regular graphs Hq (n, d) (with d × n matrices over GF(q) as vertex set, and two vertices A, B being adjacent if and only if the rank of A − B is 1) are characterized by their parameters together with the property that the number of edges of the induced subgraph on the common neighborhood of vertices x and y depends only on the distance between them. As a corollary, we prove that Hq (n, d) has no antipodal covering if n ⩾ 2d ⩾ 6 and q ⩾ 4.

A characterization of the association schemes of bilinear forms

: Consider a simple graph whose vertices are s-dimensional subspaces of a d-dimensional vector space V over (GF(q). Two vertices in this graph are adjacent if the corresponding s-dimensional subspaces intersect in an (s-1)-dimensional subspace. This graph will be called an (s,q,d)-projective graph. The Theorem 1 of this paper can be used to obtain a characterization of the (s,q,d)-projective graphs provided d is larger than some function of s and q. Characterization problems of Affine spaces and Polar spaces are also concerned in terms of flats of higher dimensions.

Characterization of projective incidence structures

3. When a map, drawing or chart, etc., was part of the material being photographed the photographer followed a definite method in "sectioning" the material. It is customary to begin photoing at the upper left hand corner of a large sheet and to continue photoing from left to right in equal sections with a small overlap. If necessary, sectioning is continued again — beginning below the first row and continuing on until complete.

Alan Peter Sprague

Papers

A characterization of projective and affine 3-schemes /