Construction of strongly regular graphs having an automorphism group of composite order
DOI:
https://doi.org/10.11575/cdm.v15i1.62323Abstract
In this paper we outline a method for constructing strongly regular graphs from orbit matrices admitting an automorphism group of composite order. In 2011, C. Lam and M. Behbahani introduced the concept of orbit matrices of strongly regular graphs and developed an algorithm for the construction of orbit matrices of strongly regular graphs with a presumed automorphism group of prime order, and construction of corresponding strongly regular graphs. The method of constructing strongly regular graphs developed and employed in this paper is a generalization of that developed by C. Lam and M. Behbahani. Using this method we classify SRGs with parameters (49,18,7,6) having an automorphism group of order six. Eleven of the SRGs with parameters (49,18,7,6) constructed in that way are new. We obtain an additional 385 new SRGs(49,18,7,6) by switching. Comparing the constructed graphs with previously known SRGs with these parameters, we conclude that up to isomorphism there are at least 727 SRGs with parameters (49,18,7,6). Further, we show that there are no SRGs with parameters (99,14,1,2) having an automorphism group of order six or nine, i.e. we rule out automorphism groups isomorphic to $Z_6$, $S_3$, $Z_9$, or $E_9$.
References
[2] M. Behbahani, C. Lam, Strongly regular graphs with non-trivial automorphisms, Discrete Math. 311 (2011), 132-144.
[3] M. Behbahani, C. Lam, P. R. J. Ostergard, On triple systems and strongly regular graphs, J. Combin. Theory Ser. A 119 (2012), 1414-1426.
[4] T. Beth, D. Jungnickel, H. Lenz, Design Theory Vol. I, Cambridge University Press, Cambridge, 1999.
[5] A.E. Brouwer, Parameters of Strongly Regular Graphs, Available at
http://www.win.tue.nl/~aeb/graphs/srg/srgtab51-100.html, Accessed on 1/10/2017.
[6] D. Crnkovic, M. O. Pavcevic, Some new symmetric designs with parameters (64,28,12), Discrete Math. 237 (2001), 109-118.
[7] D. Crnkovic, S. Rukavina, Construction of block designs admitting an abelian automorphism group, Metrika 62 (2005), 175-183.
[8] D. Crnkovic, S. Rukavina, M. Schmidt, A Classification of all Symmetric Block Designs of Order Nine with an Automorphism of Order Six, J. Combin. Des. 14 (2006), 301-312.
[9] The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.8.10; 2018. (https://www.gap-system.org)
[10] C.D. Godsil, B.D. McKay, Constructing cospectral graphs, Aequationes Math. 25 (1982), 257-268.
[11] C. Godsil, G. Royle, Algebraic Graph Theory, Springer-Verlag, New York, 2001.
[12] Z. Janko, Coset enumeration in groups and constructions of symmetric designs, Combinatorics '90 (Gaeta, 1990), 275{277, Ann. Discrete Math. 52, North-Holland, Amsterdam, 1992.
[13] D.V. Pasechnik, Skew-symmetric association schemes with two classes and strongly regular graphs of type $L_{2n-1}(4n-1)$, Interactions between algebra and combinatorics, Acta Appl. Math. 29 (1992), 129{138.
[14] V.D. Tonchev, Combinatorial Configurations: Designs, Codes, Graphs, Pitman Monographs and Surveys in Pure and Applied Mathematics 40, Wiley, New York, 1988.
[15] Wolfram Research, Inc., Mathematica, Version 11.2, Champaign, IL (2017).
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