Counting spanning trees in a small-world Farey graph

TL;DR

This paper derives the exact number of spanning trees in a Farey graph, a small-world network, using recursive relations and compares the results with other network types, contributing to understanding network enumeration problems.

Contribution

It introduces recursive relations for Laplacian polynomials of Farey graphs and provides the exact count of spanning trees, advancing network enumeration methods.

Findings

Exact number of spanning trees in Farey graph derived

Asymptotic growth constant characterized

Results compared with other network types

Abstract

The problem of spanning trees is closely related to various interesting problems in the area of statistical physics, but determining the number of spanning trees in general networks is computationally intractable. In this paper, we perform a study on the enumeration of spanning trees in a specific small-world network with an exponential distribution of vertex degrees, which is called a Farey graph since it is associated with the famous Farey sequence. According to the particular network structure, we provide some recursive relations governing the Laplacian characteristic polynomials of a Farey graph and its subgraphs. Then, making use of these relations obtained here, we derive the exact number of spanning trees in the Farey graph, as well as an approximate numerical solution for the asymptotic growth constant characterizing the network. Finally, we compare our results with those of different types of networks previously investigated.