  We prove a Li-Yau type eigenvalue-diameter estimate for signed graphs. That
is, the nonzero eigenvalues of the Laplacian of a non-negatively curved signed
graph are lower bounded by $1/D^2$ up to a constant, where $D$ stands for the
diameter. This leads to several interesting applications, including a volume
estimate for non-negatively curved signed graphs in terms of frustration index
and diameter, and a two-sided Li-Yau estimate for triangle-free graphs. Our
proof is built upon a combination of Chung-Lin-Yau type gradient estimate and a
new trick involving strong nodal domain walks of signed graphs. We further
discuss extensions of part of our results to nonlinear Laplacians on signed
graphs.
