Laplacians and the Cheeger Inequality for Directed Graphs
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We consider Laplacians for directed graphs and examine their eigenvalues. We introduce a notion of a circulation in a directed graph and its connection with the Rayleigh quotient. We then define a Cheeger constant and establish the Cheeger inequality for directed graphs. These relations can be used to deal with various problems that often arise in the study of non-reversible Markov chains including bounding the rate of convergence and deriving comparison theorems.
AMS Subject Classification.05C20 05C50 15A42 60J05
Keywords.eigenvalues Laplacian circulation the Cheeger inequality random walks Markov chains comparison theorems
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