A local characterization of the graphs of alternating forms

Summary

Abstract

Let Δ be the line graph of PG(n – 1, q), q > 2, Alt(n, q) be the graph of the n-dimensional alternating forms over GF(q), n ≥ 4. It is shown that every connected locally Δ graph, such that the number of common neighbours of any pair of vertices at distance two is the same as in Alt(n, q), is covered by Alt(n, q).

Introduction

There have been extensive studies in local characterization of graphs. Certain strongly regular graphs are characterized by their local structure. In this paper we shall investigate graphs which are locally a (q – l)-clique extension of the Grassmann graph over GF(q), q > 2. The Grassmann graph has as vertices all 2-spaces of an n-dimensional vector space V over GF(q). Two vertices are adjacent whenever they intersect nontrivially. The alternating forms graph Alt(n, q) is locally a (q – l)-clique extension of. In this paper, we restrict ourselves to the case μ = q²(q² + 1), i.e., the number of common neighbours of two vertices at distance 2 is always q²(q² + 1). Under the assumption μ = q²(q² + 1), Alt(4, q) is the only graph which is locally (q – l)-clique extension of with n = 4. This result follows from the classification of affine polar spaces due to Cohen and Shult [2].