Abstract
A spread of a strongly regular graph is a partition of the vertex set into cliques that meet Delsarte’s bound (also called Hoffman’s bound). Such spreads give rise to colorings meeting Hoffman’s lower bound for the chromatic number and to certain imprimitive three-class association schemes. These correspondences lead to conditions for existence. Most examples come from spreads and fans in (partial) geometries. We give other examples, including a spread in the McLaughlin graph. For strongly regular graphs related to regular two-graphs, spreads give lower bounds for the number of non-isomorphic strongly regular graphs in the switching class of the regular two-graph.
Research partially supported by NSA Research Grant MDA904-95-H-1019.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
T. Beth, D. Jungnickel and H. Lenz, Design Theory, Cambridge Univ. Press (1985).
A. E. Brouwer, A. Cohen and A. Neumaier, Distance-Regular Graphs, Springer-Verlag, (1989).
F. C. Bussemaker, S. Čobeljić, D. M. Cvetković and J. J. Seidel, Computer investigations of cubic graphs, T.H.-Report 76-WSK-01, Techn. Univ. Eindhoven, (1976).
P. J. Cameron, On groups with several doubly-transitive permutation representations, Math. Z., Vol. 128 (1972) pp. 1–14.
P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and Their Links, Cambridge Univ. Press, (1991).
Yaotsu Chang, Imprimitive symmetric association schemes of rank 4, University of Michigan Ph.D. Thesis, (August 1994).
F. De Clerck, A. Del Fra and D. Ghinelli, Pointsets in partial geometries, in: Advances in Finite Geometries and Designs (J. W. P. Hirschfeld, D. R. Hughes and J. A. Thas eds.), Oxford Univ. Press (1991) pp. 93–110.
P. Delsarte, An algebraic approach to the association schemes of coding theory, Philips Research Reports Suppl, Vol. 10(1973).
C.D. Godsil and A. D. Hensel, Distance regular covers of the complete graph, J. Combinatorial Theory B, Vol. 56 (1992) pp. 205–238.
J. M. Goethals and J. J. Seidel, The regular two-graph on 276 vertices, Discrete Math., Vol. 12 (1975) pp. 143–158.
W. H. Haemers, Eigenvalue techniques in design and graph theory, (Technical University Eindhoven, Ph.D. Thesis, 1979), Math. Centre Tract 121, Mathematical Centre, Amsterdam, 1980.
W. H. Haemers, Interlacing eigenvalues and graphs, Lin. Alg. Appl, Vol. 226–228 (1995) pp. 593–616.
W. H. Haemers, There exists no (76, 21, 2, 7) strongly regular graph, in: Finite Geometry and Combinatorics (F. De Clerck et al. eds.), Cambridge Univ. Press, (1993) pp. 175–176.
W. H. Haemers and D. G. Higman, Strongly regular graphs with strongly regular decomposition, Lin. Alg. Appl, Vol. 114/115 (1989) pp. 379–398.
W. H. Haemers, C. Parker, V. Pless and V. D. Tonchev, A design and a code invariant under the simple group Co3, J. Combin. Theory A, Vol. 62 (1993) pp. 225–233.
A. J. Hoffman, On eigenvalues and colorings of graphs, in: Graph Theory and its Applications (B. Harris ed.), Acad. Press, New York (1970) pp. 79–91.
J. H. van Lint and R. Wilson, A Course in Combinatorics, Cambridge Univ. Press, (1992).
R. Mathon, The systems of linked 2-(16, 6, 2) designs, Ars Comb., Vol. 11 (1981) pp. 131–148.
R. Mathon and A. Rosa, Tables of parameters of BIBD’s with r ≤ 41 including existence enumeration and resolvability results, Ann. Discrete Math., Vol. 26 (1985) pp. 275–308.
A. Neumaier, Strongly regular graphs with smallest eigenvalue -m, Archiv der Mathematik, Vol. 33 (1979) pp. 392–400.
S. E. Payne and J. A. Thas, Spreads and ovoids in finite generalized quadrangles, Geometriae Dedicata, Vol. 52 (1994) pp. 227–253.
E. Spence, Regular two-graphs on 36 vertices, Lin. Alg. Appl., Vol. 226–228 (1995) pp. 459–498.
D. E. Taylor, Regular two-graphs: Proc. London Math. Soc. Ser. 3, Vol. 35 (1977) pp. 257–274.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Additional information
Dedicated to Hanfried Lenz on the occasion of his 80th birthday
Rights and permissions
Copyright information
© 1996 Kluwer Academic Publishers, Boston
About this chapter
Cite this chapter
Haemers, W.H., Tonchev, V.D. (1996). Spreads in Strongly Regular Graphs. In: Jungnickel, D. (eds) Designs and Finite Geometries. Springer, Boston, MA. https://doi.org/10.1007/978-1-4613-1395-3_10
Download citation
DOI: https://doi.org/10.1007/978-1-4613-1395-3_10
Publisher Name: Springer, Boston, MA
Print ISBN: 978-1-4612-8604-2
Online ISBN: 978-1-4613-1395-3
eBook Packages: Springer Book Archive