A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2)

Summary

Abstract

Let Δ be the line graph of PG(n –1,2), Alt(n,2) be the graph of the n-dimensional alternating forms over GF(2), n ≥ 4. Let Γ be a connected locally Δ graph such that

1. the number of common neighbours of any pair of vertices at distance two is the same as in Alt(n,2).

2. the valency of the subgraph induced on the second neighbourhood of any vertex is the same as in Alt(n,2).

It is shown that Γ is covered either by Alt(n,2) or by the graph of (n – l)-dimensional GF(2)-quadratic forms Quad(n – 1,2).

Introduction

In this paper we investigate graphs which are locally the same as the graph Alt(n, 2) of alternating forms on an n-dimensional vector space V over GF(2). An analogous question for GF(q), q > 2, is considered in [4]. The local graph of Alt(n, 2) is isomorphic to the Grassmann graph, i.e. the line graph of PG(n–1,2). Thus, we investigate graphs which are locally. It turns out that, besides Alt(n, 2), there is another well-known graph which is locally. It is the graph Quad(n – 1,2) of quadratic forms on an (n – l)-dimensional vector space over GF(2). Both Alt(n, 2) and Quad(n – 1,2) are distance regular and have the same parameters, though they are non-isomorphic if n > 5.

Consider a half dual polar space of type D_n over GF(2). Then the collinearity graph, induced by the complement of a geometric hyperplane, is always locally.