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Suppose $G=(V,E)$ is a connected d-regular graph, possibly with loops and multiple edges. I'm interested in finding a lower bound for the smallest non-zero eigenvalue of the Laplacian in this setting. Therefore I was wondering:

Is there a generalization of Cheeger's inequality for non-simple graphs, relating the conductance to the size of the smallest non-zero eigenvalue?

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As far as I know, Cheeger's inequality does hold for non-simple graphs. If you look at Chung's book 'Spectral graph theory' she first treats simple graphs and then weighted graphs, which are even more general than loops and multiple edges. Her proof seems to hold for weighted graphs as well.

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