The Second Term in the Asymptotics for the Number of Points Moving Along a Metric Graph

We consider the problem of determining the asymptotics for the number of points moving along a metric graph. This problem is motivated by the problem of the evolution of wave packets, which at the initial moment of time are localized in a small neighborhood of one point. It turns out that the number of points, as a function of time, allows a polynomial approximation. This polynomial is expressed via Barnes’ multiple Bernoulli polynomials, which are related to the problem of counting the number of lattice points in expanding simplexes.
In this paper we give explicit formulas for the first two terms of the expansion for the counting function of the number of moving points. The leading term was found earlier and depends only on the number of vertices, the number of edges and the lengths of the edges. The second term in the expansion shows what happens to the graph when one or two edges are removed. In particular, whether it breaks up into several connected components or not. In this paper, examples of the calculation of the leading and second terms are given.