Constructing directed strongly regular graphs by using semidirect products and semidihedral groups

Abstract

In this paper, directed strongly regular graphs (DSRGs) are constructed by using semidirect products. The orbit condition in [3] has been weakened and this gives rise to the construction of DSRGs. Moreover, a different construction is given for DSRG by using semidihedral groups.

Keywords

• Directed strongly regular graphs
• semidirect products
• semidihedral groups
• Cayley graphs

References

1. Brouwer, A. E., Hobart, S., Parameters of directed strongly regular graphs. http://homepages.cwi.nl/ aeb/math/dsrg/dsrg.html
2. Duval, A., A directed graph version of strongly regular graphs, Journal of Combinatorial Theory Series A, 47 (1988), 71-100. https://doi.org/10.1016/0097-3165(88)90043-X
3. Duval, A., Iourinski, D., Semidirect product constructions of directed strongly regular graphs, Journal of Combinatorial Theory (A), 104 (2003), 157-167. https://doi.org/10.1016/S0097-3165(03)00141-9
4. Fiedler, F., Klin, M., Muzychuk, M., Small vertex-transitive directed strongly regular graphs, Discrete Mathematics, 255 (2002), 87-115. https://doi.org/10.1016/S0012-365X(01)00391-0
5. Fiedler, F., Klin, M., Pech, Ch., Directed Strongly Regular Graphs as Elements of Coherent Algebras, in: Denecke, K., Vogel, H.-J., (Eds.), General Algebra and Discrete Mathematics, Shaker Verlag, Aachen, 1999, 69-87.
6. Godsil, C. D., Hobart, S. A., Martin, W. J., Representations of directed strongly regular graphs, European J. Combin., 28(7) (2007), 1980-1993. https://doi.org/10.1016/j.ejc.2006.08.008
7. Hobart, S., Shaw, T., A note on a family of directed strongly regular graphs, European Journal of Combinatorics, 20 (1999), 819-820.
8. Jorgensen, L., Directed strongly regular graphs with $\mu=\lambda$, Discrete Mathematics, 231(1-3) (2001), 289-293.

9. Jorgensen, L., Non-existence of directed strongly regular graphs, Discrete Mathematics, 264 (2003), 111-126.
10. Klin, M., Munemasa, A., Muzychuk, M., Zieschang, P. H., Directed strongly regular graphs obtained from coherent algebras, Linear Algebra Appl., 377 (2004), 83-109.
11. Cameron, P. J., van Lint, J. H., Designs, Graphs, Codes and Their Links, Cambridge University Press, Cambridge, 1991.