Annals of Combinatorics

, Volume 9, Issue 1, pp 1–19 | Cite as

Laplacians and the Cheeger Inequality for Directed Graphs

  • Fan ChungEmail author
Original Paper


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 


eigenvalues Laplacian circulation the Cheeger inequality random walks Markov chains comparison theorems 


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Copyright information

© Birkhäuser Verlag, Basel 2005

Authors and Affiliations

  1. 1.University of CaliforniaSan Diego, La JollaUSA

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