  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
