The Number of Spanning Trees in Self-Similar Graphs

The number of spanning trees of a graph, also known as the complexity, is computed for graphs constructed by a replacement procedure yielding a self-similar structure. It is shown that under certain symmetry conditions exact formulas for the complexity can be given. These formulas indicate interesting connections to the theory of electrical networks. Examples include the well-known Sierpiński graphs and their higher-dimensional analogues. Several auxiliary results are provided on the way—for instance, a property of the number of rooted spanning forests is proven for graphs with a high degree of symmetry.