# Cheeger's inequality for graphs with multiple edges and loops?

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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