Summary
In this chapter we introduce the notion of total graph coherent configuration, and use computer tools to investigate it for two classes of strongly regular graphs – the triangular graphs T(n) and the lattice square graphs L 2(n). For T(n), we show that its total graph coherent configuration has exceptional mergings only in the cases n=5 and n=7.
This is a preview of subscription content, log in via an institution to check access.
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
L. Babel, I. V. Chuvaeva, M. Klin, and D. V. Pasechnik, Algebraic Combinatorics in Mathematical Chemistry. Methods and Algorithms. II. Program Implementation of the Weisfeiler-Leman Algorithm, Preprint, Report TUM-M9701, 1997, Fakultät für Mathematik, TU Münc. http://www-lit.ma.tum.de/veroeff/html/960.68019.html.
L. Babel, S. Baumann, and M. Luedecke, STABCOL: An Efficient Implementation of the Weisfeiler-Leman Algorithm, Technical report, Technical University Munich, TUM-M9611, 1996.
M. Behzad, G. Chartrand, and E. A. Nordhaus, Triangles in line-graphs and total graphs, Indian J. Math., 10 (1968), 109–120.
P. J. Cameron and J. H. van Lint, Designs, Graphs, Codes and Their Links, London Math. Soc. Student Texts, Vol. 22, Cambridge University Press, Cambridge, 1991.
S. Evdokimov and I. Ponomarenko, On highly closed cellular algebras and highly closed isomorphisms, Electr. J. Comb., 6 (1999), #R18.
I. A. Faradžev and M. H. Klin, Computer package for computations with coherent configurations, in Proc. ISSAC-91, pp. 219–223, ACM, Bonn, 1991.
I. A. Faradžev, M. H. Klin, and M. E. Muzichuk, Cellular rings and groups of automorphisms of graphs, in I. A. Faradžev et al. (eds.) Investigations in Algebraic Theory of Combinatorial Objects, pp. 1–152, Kluwer Academic, Dordrecht, 1994.
F. Harary, Graph Theory, Addison-Wesley, Reading, 1969.
D. G. Higman, Coherent configurations, I. Rend. Sem. Mat. Univ. Padova, 44 (1970), 1–25.
M. Klin, C. Rücker, G. Rücker, and G. Tinhofer, Algebraic combinatorics in mathematical chemistry. Methods and algorithms. I. Permutation groups and coherent (cellular) algebras, MATCH, 40 (1999), 7–138.
M. Klin, M. Ziv-Av, and L. Jorgensen, Small rank 5 Higmanian association schemes and total graph coherent configurations, in preparation.
M. Klin and M. Ziv-Av, A family of Higmanian association schemes on 40 points: A computer algebra approach, in Proceedings of an International Conference in Honor of Eiichi Bannai’s 60th Birthday, June 26–30, 2006, Algebraic Combinatorics, Sendai International Center, Sendai, Japan, pp. 190–203.
D. A. Leonard, Using Gröbner bases to investigate flag algebras and association scheme fusion, in M. Klin, G. A. Jones, A. Jurišić, M. Muzychuk, and I. Ponomarenko (eds.) Algorithmic Algebraic Combinatorics and Gröbner Bases, pp. 113–135, Springer, Berlin, 2009 (this volume).
M. Schönert, et al., GAP – Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, fifth edition, Rheinisch-Westfälische Technische Hochschule, Aachen, Germany, 1995.
B. Weisfeiler, On Construction and Identification of Graphs, Lecture Notes in Math., Vol. 558, Springer, Berlin, 1976.
H. W. Wielandt, Permutation Groups Through Invariant Relations and Invariant Functions, Lecture Notes, Ohio State University Press, Columbus, 1969.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Ziv-Av, M. (2009). Computer Aided Investigation of Total Graph Coherent Configurations for Two Infinite Families of Classical Strongly Regular Graphs. In: Klin, M., Jones, G.A., Jurišić, A., Muzychuk, M., Ponomarenko, I. (eds) Algorithmic Algebraic Combinatorics and Gröbner Bases. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-01960-9_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-01960-9_12
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-01959-3
Online ISBN: 978-3-642-01960-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)