L-RCM: a method to detect connected components in undirected graphs by using the Laplacian matrix and the RCM algorithm

In this paper we consider undirected graphs with no loops and multiple edges, consisting of k connected components. In these cases, it is well known that one can find a numbering of the vertices such that the adjacency matrix A is block diagonal with k blocks. This also holds for the (unnormalized) Laplacian matrix L= D-A, with D a diagonal matrix with the degrees of the nodes. In this paper we propose to use the Reverse Cuthill-McKee (RCM) algorithm to obtain a block diagonal form of L that reveals the number of connected components of the graph. We present some theoretical results about the irreducibility of the Laplacian matrix ordered by the RCM algorithm. As a practical application we present a very efficient method to detect connected components with a computational cost of O(m+n), being m the number of edges and n the number of nodes. The RCM method is implemented in some comercial packages like MATLAB and Mathematica. We make the computations by using the function symrcm of MATLAB. Some numerical results are shown

[1]  Jason J. Molitierno.  The Spectral Radius of Submatrices of Laplacian Matrices for Graphs with Cut Vertices , 2008 .

[2]  André Carlos Ponce de Leon Ferreira de Carvalho,et al.  Spectral methods for graph clustering - A survey , 2011, Eur. J. Oper. Res..

[3]  Jennifer A. Scott,et al.  Reducing the Total Bandwidth of a Sparse Unsymmetric Matrix , 2006, SIAM J. Matrix Anal. Appl..

[4]  E. Cuthill,et al.  Reducing the bandwidth of sparse symmetric matrices , 1969, ACM '69.

[5]  Andrew Lumsdaine,et al.  A comparison of vertex ordering algorithms for large graph visualization , 2007, 2007 6th International Asia-Pacific Symposium on Visualization.

[6]  Bojan Mohar,et al.  Laplace eigenvalues and bandwidth-type invariants of graphs , 1993, J. Graph Theory.

[7]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[8]  Alex Pothen,et al.  PARTITIONING SPARSE MATRICES WITH EIGENVECTORS OF GRAPHS* , 1990 .

[9]  Dieter Jungnickel,et al.  Graphs, Networks, and Algorithms , 1980 .

[10]  Santo Fortunato,et al.  Community detection in graphs , 2009, ArXiv.

[11]  Zhi-Li Zhang,et al.  Commute Times for a Directed Graph using an Asymmetric Laplacian , 2011 .

[12]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[13]  N. Abreu.  Old and new results on algebraic connectivity of graphs , 2007 .

[14]  A. Pothen,et al.  Two improved algorithms for envelope and wavefront reduction , 1997 .

[15]  M. Fiedler.  A property of eigenvectors of nonnegative symmetric matrices and its application to graph theory , 1975 .

[16]  A. Frommer,et al.  Algebraic multigrid methods for Laplacians of graphs , 2011 .

[17]  B. Mohar.  THE LAPLACIAN SPECTRUM OF GRAPHS y , 1991 .

[18]  A. H. Sherman,et al.  Comparative Analysis of the Cuthill–McKee and the Reverse Cuthill–McKee Ordering Algorithms for Sparse Matrices , 1976 .

[19]  Mark Newman,et al.  Networks: An Introduction , 2010 .

[20]  J. A. George.  Computer implementation of the finite element method , 1971 .

[21]  J. Pasciak,et al.  Computer solution of large sparse positive definite systems , 1982 .

[22]  Michele Benzi,et al.  Orderings for Incomplete Factorization Preconditioning of Nonsymmetric Problems , 1999, SIAM J. Sci. Comput..

[23]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[24]  Francesco Romani,et al.  Heuristic Spectral Techniques for the Reduction of Bandwidth and Work-Bound of Sparse Matrices , 2001, Numerical Algorithms.

[25]  J. Gillis,et al.  Matrix Iterative Analysis , 1961 .

[26]  John R. Gilbert,et al.  Sparse Matrices in MATLAB: Design and Implementation , 1992, SIAM J. Matrix Anal. Appl..

[27]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[28]  Robert E. Tarjan,et al.  Depth-First Search and Linear Graph Algorithms , 1972, SIAM J. Comput..

[29]  Jonathan L. Gross,et al.  Handbook of graph theory , 2007, Discrete mathematics and its applications.