The Number of Spanning Trees in Apollonian Networks

TL;DR

This paper derives an exact formula for counting spanning trees in Apollonian networks, revealing insights into their complex topological and dynamic properties relevant to various network phenomena.

Contribution

It provides the first exact analytical expression for the number of spanning trees in Apollonian networks, linking this to their unique structural features.

Findings

Exact formula for spanning trees in Apollonian networks

Calculation of spanning tree entropy for these networks

Comparison with other graphs of similar average degree

Abstract

In this paper we find an exact analytical expression for the number of spanning trees in Apollonian networks. This parameter can be related to significant topological and dynamic properties of the networks, including percolation, epidemic spreading, synchronization, and random walks. As Apollonian networks constitute an interesting family of maximal planar graphs which are simultaneously small-world, scale-free, Euclidean and space filling and highly clustered, the study of their spanning trees is of particular relevance. Our results allow also the calculation of the spanning tree entropy of Apollonian networks, which then we compare with those of other graphs with the same average degree.