#### Vol. 312, No. 1, 2021

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The limit of first eigenfunctions of the $p$-Laplacian on graphs

### Huabin Ge, Bobo Hua and Wenfeng Jiang

Vol. 312 (2021), No. 1, 103–112
##### Abstract

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.

##### Keywords
$p$-Laplacian, graph, isoperimetric inequalities
Primary: 35R02
Secondary: 35P30