Abstract. It is shown explicitly how self-similar graphs can be obtained as 'blow-up' constructions of finite cell graphsĈ. This yields a larger family of graphs than the graphs obtained by discretising continuous self-similar fractals.For a class of symmetrically self-similar graphs we study the simple random walk on a cell graphĈ, starting at a vertex v of the boundary ofĈ. It is proved that the expected number of returns to v before hitting another vertex in the boundary coincides with the resistance scaling factor.Using techniques from complex rational iteration and singularity analysis for Green functions, we compute the asymptotic behaviour of the n-step transition probabilities of the simple random walk on the whole graph. The results of Grabner and Woess for the Sierpiński graph are generalised to the class of symmetrically self-similar graphs, and at the same time the error term of the asymptotic expression is improved. Finally, we present a criterion for the occurrence of oscillating phenomena of the n-step transition probabilities.

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Abstract. The number of spanning trees of a graph, also known as the complexity, is investigated for graphs which are constructed by a replacement procedure yielding a self-similar structure. It is shown that exact formulae for the number of spanning trees can be given for sequences of self-similar graphs under certain symmetry conditions. These formulae exhibit interesting connections to the theory of electrical networks. Examples include the well-known Sierpiński graphs and their higher-dimensional analoga. Several remarkable 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.