On Locally Projective Graphs of Girth 5
A.A. Ivanov1
and Cheryl E. Praeger2
1Imperial College Department of Mathematics 180, Queen's Gate London SW7, 2BZ UK
2University of Western Australia Department of Mathematics Perth W.A 6907 Australia
2University of Western Australia Department of Mathematics Perth W.A 6907 Australia
DOI: 10.1023/A:1008619222465
Abstract
Let G @ M 23, \text or q = 4, PSL n (4) \leqslant G( x) \leqslant PGL n (4) G \cong M_{23,} {\text{or }}q = 4,PSL_n (4) \leqslant G(x) \leqslant PGL_n (4) , and 
contains the Wells graph on 32 vertices as a subgraph. In the latter case if, for a given n, at least one graph satisfying the conditions exists then there is a universal graph W(n) of which all other graphs for this n are quotients. The graph W(3) satisfies the conditions and has 2 20 vertices.
contains the Wells graph on 32 vertices as a subgraph. In the latter case if, for a given n, at least one graph satisfying the conditions exists then there is a universal graph W(n) of which all other graphs for this n are quotients. The graph W(3) satisfies the conditions and has 2 20 vertices.Pages: 259–283
Keywords: locally projective graph; graph of girth 5; 2-arc-transitive graph
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References
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11. A.A. Ivanov, “Graphs of girth 5 and diagram geometries related to the Petersen graph,” Soviet Math. Dokl. 36 (1988), 83-87.
12. A.A. Ivanov, “Graphs with projective subconstituents which contain short cycles,” in Surveys in Combinatorics, K.Walker (Ed.), London Math. Soc. Lect. Notes, Vol. 187, pp. 173-190, Cambridge Univ. Press, Cambridge 1993.
13. A.A. Ivanov and S.V. Shpectorov, “The P-geometry for M23 has no non-trivial coverings,” Europ. J. Combin. 11 (1990), 373-390.
14. A.A. Ivanov and C.E. Praeger, “On finite affine 2-arc transitive graphs,” Europ. J. Combin. 14 (1993), 421-444.
15. A.A. Ivanov and S.V. Shpectorov, “Flag-transitive tilde and Petersen type geometries are all known,” Bull. Amer. Math. Soc. 31 (1994), 173-184.
16. Ch. Jansen, K. Lux, R. Parker, and R. Wilson, An Atlas of Brauer Characters, Clarendon Press, Oxford, 1995.
17. A. Pasini, Geometries and Diagrams, Clarendon Press, Oxford, 1994.
18. M. Perkel, “Bounding the valency of polygonal graphs with small girth,” Canad. J. Math. 31 (1979), 1307- 1321.
19. V.I. Trofimov, “Stabilizers of vertices of graphs with projective suborbits,” Soviet Math. Dokl. 42 (1991), 825-828.
20. R. Weiss, “s-Transitive graphs,” Algebraic Methods in Graph Theory, North-Holland, Amsterdam, 1981, pp. 827-847.
21. A.L. Wells, “Regular generalized switching classes and related topics,” D. Phil. Thesis, University of Oxford, 1983.
2. C. Armanios, “A new 5-valent distance transitive graph,” Ars Combin. 19A (1985), 77-85.
3. G. Bell, “On the cohomology of the finite special linear groups I,” J. Algebra 54 (1978), 216-238.
4. A. Brouwer, A. Cohen, and A. Neumaier, Distance Regular Graphs, Springer, Berlin, 1989.
5. P.J. Cameron and C.E. Praeger, “Graphs and permutation groups with projective subconstituents,” J. London Math. Soc. 25 (1982), 62-74.
6. K. Ching, “Graphs of small girth which are locally projective spaces,” Ph.D. Thesis, Tufts University, Medford, MA, 1992.
7. J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, and R.A. Wilson, Atlas of Finite Groups, Clarendon Press, Oxford,
1985. P1: ICA Journal of Algebraic Combinatorics KL559-03-Ivanov February 18, 1998 11:32 LOCALLY PROJECTIVE GRAPHS OF GIRTH 5 283
8. L.E. Dickson, Linear Groups with an Exposition of the Galois Field Theory, Dover Edition, Dover, New York, 1958.
9. D.R. Hughes, “Extensions of designs and groups: projective, symplectic and certain affine groups,” Math. Z. 89 (1965), 199-205.
10. A.A. Ivanov, “On 2-transitive graphs of girth 5,” Europ. J. Combin. 8 (1987), 393-420.
11. A.A. Ivanov, “Graphs of girth 5 and diagram geometries related to the Petersen graph,” Soviet Math. Dokl. 36 (1988), 83-87.
12. A.A. Ivanov, “Graphs with projective subconstituents which contain short cycles,” in Surveys in Combinatorics, K.Walker (Ed.), London Math. Soc. Lect. Notes, Vol. 187, pp. 173-190, Cambridge Univ. Press, Cambridge 1993.
13. A.A. Ivanov and S.V. Shpectorov, “The P-geometry for M23 has no non-trivial coverings,” Europ. J. Combin. 11 (1990), 373-390.
14. A.A. Ivanov and C.E. Praeger, “On finite affine 2-arc transitive graphs,” Europ. J. Combin. 14 (1993), 421-444.
15. A.A. Ivanov and S.V. Shpectorov, “Flag-transitive tilde and Petersen type geometries are all known,” Bull. Amer. Math. Soc. 31 (1994), 173-184.
16. Ch. Jansen, K. Lux, R. Parker, and R. Wilson, An Atlas of Brauer Characters, Clarendon Press, Oxford, 1995.
17. A. Pasini, Geometries and Diagrams, Clarendon Press, Oxford, 1994.
18. M. Perkel, “Bounding the valency of polygonal graphs with small girth,” Canad. J. Math. 31 (1979), 1307- 1321.
19. V.I. Trofimov, “Stabilizers of vertices of graphs with projective suborbits,” Soviet Math. Dokl. 42 (1991), 825-828.
20. R. Weiss, “s-Transitive graphs,” Algebraic Methods in Graph Theory, North-Holland, Amsterdam, 1981, pp. 827-847.
21. A.L. Wells, “Regular generalized switching classes and related topics,” D. Phil. Thesis, University of Oxford, 1983.