Abstract
In this paper we obtain an intermediate value theorem for the decycling numbers of Toeplitz graphs: if \(n \ge 3\), and \(0 \le r \le n - 2\), then there exists a Toeplitz graph of order \(n\) with decycling number \(r\). We also prove that the decycling numbers of connected Cayley graphs of order \(n\) satisfy the intermediate value property if and only if \(n = 4\) or \(6\).
Similar content being viewed by others
References
Bau, S.: A generalization of the concept of Toeplitz graphs. Mong. Math. J. 15, 54–61 (2011)
Bau, S., Beineke, L.W.: The decycling number of graphs. Austral. J. Combin. 25, 285–298 (2002)
Beineke, L.W., Vandell, R.C.: Decycling graphs. J. Gr. Theory 25, 59–77 (1996)
Gray, R.M.: Toeplitz and circulant matrices: a review, http://ee.stanford.edu/~gray/toeplitz.pdf. Accessed 25 Nov 2014
Gray, R.M.: On the eigenvalue distribution of Toeplitz graphs. IEEE Trans. Inf. Theory IT–16, 412–421 (1970)
Halmos, P.: A Hilbert Space Problem Book. American Book Company, USA (1967)
Punnim, N.: Switchings, realizations, and interpolation theorems for graph parameters. Int. J. Math. Math. Sci. 13, 2095–2117 (2005)
van Dal, R., Tijssen, G., Tuza, Z., van der Veen, A.A., Zamfirescu, Ch., Zamfirescu, T.: Hamiltonian properties of Toeplitz graphs. Discret. Math. 159, 69–81 (1996)
Zelinka, B.: Graphs of semigroups. Cas. Pest. Mat. 106, 407–408 (1981)
Zhou, S.-M.: Interpolation theorems for graphs, hypergraphs and matroids. Discret. Math. 185, 221–229 (1998)
Acknowledgments
The authors thank the referees whose comments improved the presentation of the paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Research supported by NRF South Africa.
Rights and permissions
About this article
Cite this article
Bau, S., van Niekerk, B. & White, D. An Intermediate Value Theorem for the Decycling Numbers of Toeplitz Graphs. Graphs and Combinatorics 31, 2037–2042 (2015). https://doi.org/10.1007/s00373-014-1492-3
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-014-1492-3