How the permutation of edges of a metric graph affects the number of   points moving along the edges
  We consider a dynamical system on a metric graph, that corresponds to a semiclassical solution of a time-dependent Schr\"odinger equation. We omit all details concerning mathematical physics and work with a purely discrete problem. We find a weak inequality representation for the number of points coming out of the vertex of an arbitrary tree graph. We apply this construction to an "H-junction" graph. We calculate the difference between numbers of moving points corresponding to the permutation of edges. Then we find a symmetrical difference of the number of points moving along the edges of a metric graph. 
V.L. Chernyshev, A.A. Tolchennikov
math-ph math.MP