The number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs

• We find analytically the number of spanning trees of a relevant complex network model.
• Spanning tree entropy is the lowest reported for lattices with the same average degree.
• The counting method relies on self-similarity and can be adapted for other graphs.
• Graphs are outerplanar and many NP-complete algorithms run in polynomial time on them.

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.