Statistics for Impulses on a Metric Graph for the Case of Rationally Dependent Travel Times of Edges

In this paper we consider a dynamic system on a metric graph with edges of rational length. The impulses (points) propagates through the graph. When several impulses come to the vertex of degree q at the same time, the impulses travel through all the q edges incident to this vertex. The main goal was to find the time of stabilization for each of the types of considered graphs using computer experiments and propose hypotheses about how it is expressed through the characteristics of the graph. Hypotheses were subsequently either confirmed and proven, or disproved. The paper considered graphs with edges of the same length, star graphs, graphs with two common to all edges vertices, and a complete graph with three vertices. As a result, for the majority of cases formulas for the stabilization time were obtained and proved. It turned out that they are related to the concept of eccentricity for the initial vertex, the presence of an odd length in the graph of the cycle and the Frobenius number for the edge lengths. In addition, the paper considered a matrix approach to the problem of finding the stabilization time, in which the problem for a metric graph is reduced to a problem on another oriented discrete graph. It was shown that in the standard form this approach is not very effective. Further progress can be associated with the further usage of combinatorial methods and with the application of the matrix approach with proper modification. Keywords: metric graph, stabilization time, eccentricity of the vertex of the graph, cycle of odd length, Frobenius number, adjacency matrix of the oriented graph.