Abstract：In this paper, we study eigenvalues and eigenfunctions of p-Laplacians with Dirichlet boundary condition on graphs. We characterize the ?rst eigenfunction (and the maximum eigenfunction for a bipartite graph) via the sign condition. By the uniqueness of the ?rst eigenfunction of p-Laplacian, as p → 1, we identify the Cheeger constant of a symmetric graph with that of the quotient graph. By this approach, we calculate various Cheeger constants of spherically symmetric graphs. This is a joint work with B. Hua.