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RECOGNITION OF STRONGLY CONNECTED COMPONENTS BY THE LOCATION OF NONZERO ELEMENTS OCCURRING IN C(G) = (D - A(G))-1

  • Kim, Koon-Chan (Department of Mathematics, Keimyung University) ;
  • Kang, Young-Yug (Department of Mathematics, Keimyung University)
  • Published : 2004.02.01

Abstract

One of the intriguing and fundamental algorithmic graph problems is the computation of the strongly connected components of a directed graph G. In this paper we first introduce a simple procedure for determining the location of the nonzero elements occurring in $B^{-1}$ without fully inverting B, where EB\;{\equiv}\;(b_{ij)\;and\;B^T$ are diagonally dominant matrices with $b_{ii}\;>\;0$ for all i and $b_{ij}\;{\leq}\;0$, for $i\;{\neq}\;j$, and then, as an application, show that all of the strongly connected components of a directed graph G can be recognized by the location of the nonzero elements occurring in the matrix $C(G)\;=\;(D\;-\;A(G))^{-1}$. Here A(G) is an adjacency matrix of G and D is an arbitrary scalar matrix such that (D - A(G)) becomes a diagonally dominant matrix.

Keywords

strongly connected components;directed graph;inverse matrix;diagonally dominant matrix

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