The Number of Spanning Trees of an Infinite Family of Outerplanar, Small-World and Self-Similar Graphs

TL;DR

This paper derives an exact formula for the number of spanning trees in an infinite family of outerplanar, small-world, and self-similar graphs, revealing insights into their topological and dynamic properties.

Contribution

It provides the first closed-form analytical expression for the spanning trees of this specific class of infinite graphs, including their entropy.

Findings

Exact formula for spanning trees of the graph family

Comparison of spanning tree entropy with similar average degree graphs

Insights into graph reliability and diffusion properties

Abstract

In this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. This number is an important graph invariant related to different topological and dynamic properties of the graph, such as its reliability, synchronization capability and diffusion properties. The calculation of the number of spanning trees is a demanding and difficult task, in particular for large graphs, and thus there is much interest in obtaining closed expressions for relevant infinite graph families. We have also calculated the spanning tree entropy of the graphs which we have compared with those for graphs with the same average degree.