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

TL;DR

This paper introduces a novel method using the RCM algorithm on the Laplacian matrix to efficiently identify connected components in undirected graphs, with theoretical insights and practical implementation details.

Contribution

The paper proposes a new approach combining the RCM algorithm and Laplacian matrix to detect connected components efficiently, with theoretical analysis and practical computational benefits.

Findings

The method correctly identifies connected components in linear time.

The approach is validated with numerical experiments and implemented in MATLAB.

Theoretical results support the irreducibility of the RCM-ordered Laplacian.

Abstract

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