Paths counting on simple graphs: from escape to localization

We study the asymptotic behavior of the number of paths of length $N$ on several classes of infinite graphs with a single special vertex. This vertex can work as an entropic trap for the path, i.e. under certain conditions the dominant part of long paths become localized in the vicinity of the special point instead of spreading to infinity. We study the conditions for such localization on decorated star graphs, regular trees and regular hyperbolic graphs as a function of the functionality of the special vertex. In all cases the localization occurs for large enough functionality. The particular value of transition point depends on the large-scale topology of the graph. The emergence of localization is supported by the analysis of the spectra of the adjacency matrices of corresponding finite graphs.