Compact form of the pseudo-inverse matrix in the approximation of a star graph using the conductance electrical model (CEM)

Abstract

The Conductance Electrical Model (CEM) transforms a graph into a circuit and can be use to do inexact graph isomorphism as it was shown in [13]. In second stage of this process, we transform the circuit r eq in a star circuit, using the Moore-Penrose pseudo-inverse of a matrix for which there is a general formula that requires transpose, multiply and invert matrices with a time complexity of O(N 4), where N is the number of nodes of the graph. However, due to the special structure of the star transformation, we are able to exploit this special structure to compute the pseudo-inverse without using the general Moore-Penrose formula. We have developed a closed formula that can compute the elements of the pseudo-inverse without using that formula, that means without multiplying matrices neither doing the matrix inversion and that moreover can be computed in O(N 3). This method also eliminates the problems due to computer rounding and due to bad-conditioned problems in mathematical terms. © 2012 Springer-Verlag Berlin Heidelberg.