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

Title & Authors
RECOGNITION OF STRONGLY CONNECTED COMPONENTS BY THE LOCATION OF NONZERO ELEMENTS OCCURRING IN C(G) = (D - A(G))-1
Kim, Koon-Chan; Kang, Young-Yug;

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 $\small{B^{-1}}$ without fully inverting B, where $\small{EB\;{\equiv}\;(b_{ij)\;and\;B^T}$ are diagonally dominant matrices with $b_{ii}\;>\;0$ for all i and $\small{b_{ij}\;{\leq}\;0}$, for $\small{i\;{\neq}\;j}$, 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 $\small{C(G)\;=\;(D\;-\;A(G))^{-1}}$. 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;
Language
English
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