The number and degree distribution of spanning trees in the Tower of Hanoi graph

TL;DR

This paper analytically determines the number and degree distribution of spanning trees in the Hanoi graph, providing insights into its structural properties and comparing its spanning tree entropy with similar graphs.

Contribution

It introduces recursion relations and an exact formula for counting spanning trees in the Hanoi graph, and develops a vertex labeling method to analyze degree distributions.

Findings

Exact number of spanning trees for the Hanoi graph

Spanning tree entropy comparison with similar graphs

Vertex labeling method for degree distribution analysis

Abstract

The number of spanning trees of a graph is an important invariant related to topological and dynamic properties of the graph, such as its reliability, communication aspects, synchronization, and so on. However, the practical enumeration of spanning trees and the study of their properties remain a challenge, particularly for large networks. In this paper, we study the num- ber and degree distribution of the spanning trees in the Hanoi graph. We first establish recursion relations between the number of spanning trees and other spanning subgraphs of the Hanoi graph, from which we find an exact analytical expression for the number of spanning trees of the n-disc Hanoi graph. This result allows the calculation of the spanning tree entropy which is then compared with those for other graphs with the same average degree. Then, we introduce a vertex labeling which allows to find, for each vertex of the graph, its degree distribution among all possible spanning trees.