Discrete Mathematics & Theoretical Computer Science, Vol 11, No 1 (2009)

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Asymptotic enumeration on self-similar graphs with two boundary vertices

Elmar Teufl, Stephan Wagner


We study two graph parameters, namely the number of spanning forests and the number of connected subgraphs, for self-similar graphs with exactly two boundary vertices. In both cases, we determine the general behavior for these and related auxiliary quantities by means of polynomial recurrences and a careful asymptotic analysis. It turns out that the so-called resistance scaling factor of a graph plays an essential role in both instances, a phenomenon that was previously observed for the number of spanning trees. Several explicit examples show that our findings are likely to hold in an even more general setting.

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