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We study the limit of first eigenfunctions of (discrete) $p$-Laplacian on afinite subset of a graph with Dirichlet boundary condition, as $p\to 1.$ Weprove that up to a subsequence, they converge to a summation of characteristicfunctions of Cheeger cuts of the graph. We give an example to show that thelimit may not be a characteristic function of a single Cheeger cut.

A note on limit of first eigenfunctions of $p$-Laplacian on graphs