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Generalized Cheeger inequalities for eigenvalues of non-reversible Markov chains

We show lower bounds for the smallest non-trivial eigenvalue, and smallest real portion of an eigenvalue, of the Laplacian of a non-reversible Markov chain in terms of an Evolving set quantity. A myriad of Cheeger-like inequalities follow for non-reversible chains, which even in the reversible case sharpen previously known results. The same argument also produces a new Cheeger-like inequality for the smallest eigenvalue of a reversible chain, and a Cheeger-like inequality for the second largest magnitude eigenvalue of a non-reversible chain.