We consider graph classes G in which every graph has components in a class C of connected graphs. We provide a framework for the asymptotic study of |G n,N |, the number of graphs in G with n vertices and N := λn components, where λ ∈ (0, 1). Assuming that the number of graphs with n vertices in C satisfies
We study random unrooted plane trees with n vertices sampled according to the weights corresponding to the vertex-degrees. Our main result shows that if the generating series of the weights has positive radius of convergence, then this model of random trees may be approximated geometrically by a Galton-Watson tree conditioned on having a large random size. This implies that a variety of results for the well-studied planted case also hold for unrooted trees, including Gromov-Hausdorff-Prokhorov scaling limits, tail-bounds for the diameter, distributional graph limits, and limits for the maximum degree. Our work complements results by Wang (2016), who studied random unrooted plane trees whose diameter tends to infinity.
Consider a non-negative sequence cn = h(n) • n α−1 • ρ −n , where h is slowly varying, α > 0, 0 < ρ < 1 and n ∈ N. We investigate the coefficients of G(x, y) = k≥1 (1 − x k y) −c k , which is the bivariate generating series of the multiset construction of combinatorial objects. By a powerful blend of probabilistic methods based on the Boltzmann model and analytic techniques exploiting the well-known saddle-point method we determine the number of multisets of total size n with N components, that is, the coefficient of x n y N in G(x, y), asymptotically as n → ∞ and for all ranges of N . Our results reveal a phase transition in the structure of the counting formula that depends on the ratio n/N and that demonstrates a prototypical passage from a bivariate local limit to an univariate one.
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