On the rank of the Doob graph and its complement
Keywords:
Doob graph, Complement graph, Circulant graph, Rank, Determinant.Abstract
We compute the rank of the circulant Doob graph defined in Doob (2002). We also compute
the rank and the determinant of its complement graph.
References
Cvetković, D., Rowlinson, P. & Simić, S. (2010). Graph
operations and modifications. In: An Introduction to the
Theory of Graph Spectra, Pp. 25. London Mathematical
Society Student Texts 75. Cambridge University Press,
England.
Davis, P. J. (1979). Centralizers and circulants. In:
Circulant Matrices, Pp. 208. American Mathematical
Society Chelsea Publishing, Providence, RI.
Doob, M. (2002). Circulant graphs with det(−A(G))=−
deg(G): Codeterminants with Kn. Linear Algebra and its
Applications, 340: 87–96.
Garner, C.R. (2004). Investigations into the ranks of
regular graphs. PhD thesis, Rand Afrikaans University,
Johannesburg, RSA.
Gray, R. M. (2006). Toeplitz and circulant matrices: A
review. Foundations and Trends® in Communications
and Information Theory, 2(3): 155–239.
Sookyang, S., Arworn, S. & Wojtylak, P. (2008).
Characterizations of non-singular cycles, path and trees.
Thai Journal of Mathematics, 6(2): 331–336.
Williams, G. (2014). Smith forms for adjacency matrices
of circulant graphs. Linear Algebra and its Applications,
: 21–33.
Wyn-jones, A. (2013). Circulant matrices.
In: Circulants, Pp. 12 -21. Retrieved from
CiteSeerX. http://www.circulants.org.
