Abstract
In this paper we present some theoretical results about the irreducibility of the Laplacian matrix ordered by the Reverse Cuthill-McKee (RCM) algorithm. We consider undirected graphs with no loops consisting of some connected components. RCM is a well-known scheme for numbering the nodes of a network in such a way that the corresponding adjacency matrix has a narrow bandwidth. Inspired by some properties of the eigenvectors of a Laplacian matrix, we derive some properties based on row sums of a Laplacian matrix that was reordered by the RCM algorithm. One of the theoretical results serves as a basis for writing an easy MATLAB code to detect connected components, by using the function “symrcm” of MATLAB. Some examples illustrate the theoretical results.
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Dedicated to the memory of Miroslav Fiedler
The research has been supported by Spanish DGI grant MTM2010-18674, Consolider Ingenio CSD2007-00022, PROMETEO 2008/051, OVAMAH TIN2009-13839-C03-01, and PAID-06-11-2084.
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Pedroche, F., Rebollo, M., Carrascosa, C. et al. On some properties of the Laplacian matrix revealed by the RCM algorithm. Czech Math J 66, 603–620 (2016). https://doi.org/10.1007/s10587-016-0281-y
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DOI: https://doi.org/10.1007/s10587-016-0281-y