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
Locally co-Heawood graphs
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 incidence graph of the unique biplane on 7 points, that is, the bipartite complement of the Heawood graph. We find that there are precisely three connected graphs that are locally Δ, on 36, 48 and 108 vertices, where the last graph is an antipodal 3-cover of the first one.
Introduction
Let notation be as in [1]. (In particular, ∼ denotes adjacency, Γi(γ) is the collection of vertices at distance i from γ in Γ, Γ(γ) := Γ1(γ), and γ⊥ := {γ} ∪ Γ(γ).) The Heawood graph H is the smallest cubic graph of girth 6; it is bipartite, the incidence graph of the Fano plane. The co-Heawood graph Δ is its bipartite complement, the nonincidence graph of the Fano plane, i.e., the incidence graph of the unique biplane on 7 points. (Thus, Δ = H3.) The graph Δ has 14 vertices, valency 4, is bipartite, is distance-regular of diameter 3 and has distance distribution diagram
Its automorphism group is G ≃ PGL(2, 7) of order 336 acting distance transitively.
The graph Δ occurs in the Suzuki chain S0 = 4K1, S1 = Δ, S2, S3, S4, S5 of graphs on 4, 14, 36, 100, 416, 1782 vertices, respectively. Each graph Si+1 of this chain is locally Si. In particular, the graph Σ := S2 is locally Δ, it is strongly regular with parameters (v, k, λ, μ) = (36,14,4,6).
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- Finite Geometries and Combinatorics , pp. 59 - 68Publisher: Cambridge University PressPrint publication year: 1993
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