Abstract
The number of spanning trees in a class of directed circulant graphs with generators depending linearly on the number of vertices \(\beta n\), and in the nth and \((n-1)\)th power graphs of the \(\beta n\)-cycle are evaluated as a product of \(\lceil \beta /2\rceil -1\) terms.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Aigner, M.: A Course in Enumeration, Graduate Texts in Mathematics, vol. 238. Springer, Berlin (2007)
Atajan, T., Otsuka, N., Yong, X.: Counting the number of spanning trees in a class of double fixed-step loop networks. Appl. Math. Lett. 23(3), 291–298 (2010). doi:10.1016/j.aml.2009.04.006
Atajan, T., Yong, X., Inaba, H.: Further analysis of the number of spanning trees in circulant graphs. Discrete Math. 306(22), 2817–2827 (2006). doi:10.1016/j.disc.2006.05.024
Biggs, N.: Algebraic Graph Theory, Cambridge Mathematical Library. Cambridge University Press, Cambridge (1993)
Chen, X.: The number of spanning trees in directed circulant graphs with non-fixed jumps. Discrete Math. 307(15), 1873–1880 (2007). doi:10.1016/j.disc.2006.09.034
Golin, M.J., Yong, X., Zhang, Y.: The asymptotic number of spanning trees in circulant graphs. Discrete Math. 310(4), 792–803 (2010). doi:10.1016/j.disc.2009.09.008
Hwang, F.K.: A complementary survey on double-loop networks. Theor. Comput. Sci. 263(1–2), 211–229 (2001). doi:10.1016/S0304-3975(00)00243-7. (Combinatorics and computer science (Palaiseau, 1997))
Hwang, F.K.: A survey on multi-loop networks. Theor. Comput. Sci. 299(1–3), 107–121 (2003). doi:10.1016/S0304-3975(01)00341-3
Li, D., Liu, M., Peng, Y.: Hajós’ conjecture and cycle power graphs. Eur. J. Comb. 31(3), 759–764 (2010). doi:10.1016/j.ejc.2009.08.008
Lih, K.W., Liu, D.D.F., Zhu, X.: Star extremal circulant graphs. SIAM J. Discrete Math. 12(4), 491–499 (1999). doi:10.1137/S0895480198342838
Louis, J.: Asymptotics for the number of spanning trees in circulant graphs and degenerating d-dimensional discrete tori. Ann. Comb. 19(3), 513–543 (2015). doi:10.1007/s00026-015-0272-y
Louis, J.: A formula for the number of spanning trees in circulant graphs with nonfixed generators and discrete tori. Bull. Aust. Math. Soc. 92(3), 365–373 (2015). doi:10.1017/S0004972715000969
Yong, X., Zhang, Y., Golin, M.J.: The number of spanning trees in a class of double fixed-step loop networks. Networks 52(2), 69–77 (2008). doi:10.1002/net.20223
Acknowledgments
The author thanks Anders Karlsson for reading the manuscript and useful discussions. The author also acknowledges the anonymous referee for useful comments and for correcting a mistake in the last corollary.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by A. Constantin.
The author acknowledges support from the Swiss NSF grant 200021 132528/1.
Rights and permissions
About this article
Cite this article
Louis, J. Spanning trees in directed circulant graphs and cycle power graphs. Monatsh Math 182, 51–63 (2017). https://doi.org/10.1007/s00605-016-0912-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00605-016-0912-2