Trajectory Localization on a Finite Regular Graph with a Defect

In this work, two tree-like graph classes are studied. Both have the same vertex degree p, excluding root of degree p0 and lists. In one case the lists are not interconnected, meaning that their degrees are 1. In other case the lists are interconnected in such a way that their degrees equal p. The goal of this work is to analyze and compare asymptotic behavior of path counting on such graphs. Finite and infinite tree-like graphs with one special vertex in the root were reviewed in this study. Regular tree-like graphs with one special vertex in the root were analyzed in this study. Comparing of asymptotic behavior of the number of paths in these two classes of graphs was conducted and the formulas for relaxation time depending on vertex degree and number of generations of regular graph were derived. The Wolfram language algorithms were developed for that purpose using Wolfram Mathematica.