Abstract
For an abelian Cayley graph of degree 2n and diameter 2, it has at most \(2n^2+2n+1\) vertices which meets the abelian Cayley–Moore bound. Recently, Leung and the second author proved that such a graph exists if and only if \(n=1,2\). A natural question is that whether one can also classify abelian Cayley graphs of degree 2n and diameter 2 with exactly \(2n^2+2n\) vertices. For \(n=1,2\), there are examples. As the total number of vertices of such graphs is one smaller than the abelian Cayley–Moore bound, we call them of defect one. By using some algebraic approaches, we provide several nonexistent results for infinitely many \(n >2\).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.
References
Bannai, E., Ito, T.: On finite Moore graphs. J. Fac. Sci. Univ. Tokyo Sect. A Math. 20, 191–208 (1973)
Beth, T., Jungnickel, D., Lenz, H.: Design Theory, vol. I. Cambridge University Press, Cambridge (1999)
Bosma, W., Cannon, J., Playoust, C.: The magma algebra system I: the user language. J. Symb. Comput. 24(3–4), 235–265 (1997)
Damerell, R.M.: On Moore graphs. Math. Proc. Camb. Philos. Soc. 74(2), 227–236 (1973)
Dougherty, R., Faber, V.: The degree-diameter problem for several varieties of Cayley graphs I: the abelian case. SIAM J. Discret. Math. 17(3), 478–519 (2004)
Golomb, S.W., Welch, L.R.: Perfect codes in the Lee metric and the packing of polyominoes. SIAM J. Appl. Math. 18(2), 302–317 (1970)
Hoffman, A.J., Singleton, R.R.: On Moore graphs with diameters 2 and 3. IBM J. Res. Dev. 4(5), 497–504 (1960)
Horak, P., Kim, D.: 50 years of the Golomb-Welch conjecture. IEEE Trans. Inf. Theory 64(4), 3048–3061 (2018)
Kim, Dongryul: Nonexistence of perfect 2-error-correcting Lee codes in certain dimensions. Eur. J. Comb. 63, 1–5 (2017)
Leung, K. H., Zhou, Y.: No lattice tiling of \({\mathbb{Z}}^n\) by Lee sphere of radius 2. J. Comb. Theory Ser. A 171 (2020)
Miller, M., Širáň, J.: Moore graphs and beyond: A survey of the degree/diameter problem. Electron. J. Comb. 20(2), DS14 (2013)
Neukirch, J.: Algebraic Number Theory. Springer-Verlag, Berlin Heidelberg (1999)
Pott, A.: Finite Geometry and Character Theory. Springer Verlag, Berlin (1995)
Weiss, E.: Algebraic number theory, unabridged edn Dover Publications, Mineola (1998)
Zhang, T., Zhou, Y.: On the nonexistence of lattice tilings of \({\mathbb{Z} }^{n}\) by Lee spheres. J. Comb. Theory Ser. A 165, 225–257 (2019)
Acknowledgements
This work is supported by the Natural Science Foundation of Hunan Province (No. 2019JJ30030), the Training Program for Excellent Young Innovators of Changsha (No. kq2106006) and the Fund for NUDT Young Innovator Awards (No. 20180101).
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
He, W., Zhou, Y. On abelian cayley graphs of diameter two and defect one. J Algebr Comb 58, 137–156 (2023). https://doi.org/10.1007/s10801-023-01241-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10801-023-01241-7