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