  We establish various nodal domain theorems for $p$-Laplacians on signed
graphs, which unify most of the existing results on nodal domains of graph
$p$-Laplacians and arbitrary symmetric matrices. Based on our nodal domain
estimates, we obtain a higher order Cheeger inequality that relates the
variational eigenvalues of $p$-Laplacians and Atay-Liu's multi-way Cheeger
constants on signed graphs. In the particular case of $p=1$, this leads to
several identities relating variational eigenvalues and multi-way Cheeger
constants. Intriguingly, our approach also leads to new results on usual
graphs, including a weak version of Sturm's oscillation theorem for graph
$1$-Laplacians and nonexistence of eigenvalues between the largest and second
largest variational eigenvalues of $p$-Laplacians with $p>1$ on connected
bipartite graphs.
