As a nonlinear extension of the graph Laplacian, the graph $p$-Laplacian has various applications in many fields. Due to the nonlinearity, it is very difficult to compute the eigenvalues and eigenfunctions of graph $p$-Laplacian. In this paper, we establish the equivalence between the graph $p$-Laplacian eigenproblem and the tensor eigenproblem when $p$ is even. Building on this result, algorithms designed for tensor eigenproblems can be adapted to compute the eigenpairs of the graph $p$-Laplacian. For general $p>1$, we give a fast and convergent algorithm to compute the largest eigenvalue and the corresponding eigenfunction of the signless graph $p$-Laplacian. As an application, we provide a new criterion to determine when a graph is not a subgraph of another one, which outperforms existing criteria based on the linear Laplacian and adjacency matrices. Our work highlights the deep connections and numerous similarities between the spectral theories of tensors and graph $p$-Laplacians.