Steklov flows on trees and applications

We introduce the Steklov flows on finite trees, i.e. the flows (or currents) associated with the Steklov problem. By constructing appropriate Steklov flows, we prove the monotonicity and rigidity of the first nonzero Steklov eigenvalues on trees: for finite trees $\g_1$ and $\g_2,$ the first nonzero Steklov eigenvalue of $\g_1$ is greater than or equal to that of $\g_2$, provided that $\g_1$ is a subgraph of $\g_2.$ Moreover, we give the sufficient and necessary condition in which the equality holds.