Book contents
- Frontmatter
- Contents
- Preface
- Introduction
- Generalized hexagons and BLT-sets
- Orthogonally divergent spreads of Hermitian curves
- Lifts of nuclei in finite projective spaces
- Large minimal blocking sets, strong representative systems, and partial unitals
- The complement of a geometric hyperplane in a generalized polygon is usually connected
- Locally co-Heawood graphs
- A theorem of Parmentier characterizing projective spaces by polarities
- Geometries with diagram (diagram omitted)
- Remarks on finite generalized hexagons and octagons with a point-transitive automorphism group
- Block-transitive t-designs, II: large t
- Generalized Fischer spaces
- Ovoids and windows in finite generalized hexagons
- Flag transitive L.C2 geometries
- On nonics, ovals and codes in Desarguesian planes of even order
- Orbits of arcs in projective spaces
- There exists no (76,21,2,7) strongly regular graph
- Group-arcs of prime power order on cubic curves
- Planar Singer groups with even order multiplier groups
- On a footnote of Tits concerning Dn-geometries
- The structure of the central units of a commutative semifield plane
- Partially sharp subsets of PΓL(n, q)
- Partial ovoids and generalized hexagons
- A census of known flag-transitive extended grids
- Root lattice constructions of ovoids
- Coxeter groups in Coxeter groups
- A local characterization of the graphs of alternating forms
- A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2)
- On some locally 3-transposition graphs
- Coherent configurations derived from quasiregular points in generalized quadrangles
- Veldkamp planes
- The Lyons group has no distance-transitive representation
- Intersection of arcs and normal rational curves in spaces of odd characteristic
- Flocks and partial flocks of the quadratic cone in PG(3, q)
- Some extended generalized hexagons
- Nuclei in finite non-Desarguesian projective planes
A local characterization of the graphs of alternating forms
Published online by Cambridge University Press: 07 September 2010
- Frontmatter
- Contents
- Preface
- Introduction
- Generalized hexagons and BLT-sets
- Orthogonally divergent spreads of Hermitian curves
- Lifts of nuclei in finite projective spaces
- Large minimal blocking sets, strong representative systems, and partial unitals
- The complement of a geometric hyperplane in a generalized polygon is usually connected
- Locally co-Heawood graphs
- A theorem of Parmentier characterizing projective spaces by polarities
- Geometries with diagram (diagram omitted)
- Remarks on finite generalized hexagons and octagons with a point-transitive automorphism group
- Block-transitive t-designs, II: large t
- Generalized Fischer spaces
- Ovoids and windows in finite generalized hexagons
- Flag transitive L.C2 geometries
- On nonics, ovals and codes in Desarguesian planes of even order
- Orbits of arcs in projective spaces
- There exists no (76,21,2,7) strongly regular graph
- Group-arcs of prime power order on cubic curves
- Planar Singer groups with even order multiplier groups
- On a footnote of Tits concerning Dn-geometries
- The structure of the central units of a commutative semifield plane
- Partially sharp subsets of PΓL(n, q)
- Partial ovoids and generalized hexagons
- A census of known flag-transitive extended grids
- Root lattice constructions of ovoids
- Coxeter groups in Coxeter groups
- A local characterization of the graphs of alternating forms
- A local characterization of the graphs of alternating forms and the graphs of quadratic forms over GF(2)
- On some locally 3-transposition graphs
- Coherent configurations derived from quasiregular points in generalized quadrangles
- Veldkamp planes
- The Lyons group has no distance-transitive representation
- Intersection of arcs and normal rational curves in spaces of odd characteristic
- Flocks and partial flocks of the quadratic cone in PG(3, q)
- Some extended generalized hexagons
- Nuclei in finite non-Desarguesian projective planes
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].
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- Finite Geometries and Combinatorics , pp. 289 - 302Publisher: Cambridge University PressPrint publication year: 1993
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