The asymptotic number of acyclic digraphs. I

Abstract

We obtain an asymptotic formula forA n,q , the number of digraphs withn labeled vertices,q edges and no cycles. The derivation consists of two separate parts. In the first we analyze the generating function forA n,q so as to obtain a central limit theorem for an associated probability distribution. In the second part we show combinatorially thatA n,q is a smooth function ofq. By combining these results, we obtain the desired asymptotic formula.

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Research supported by NSF under grant MCS-8300414.

Research supported by NSERC under grant A4067.

Research supported by NSF under grant MCS-8302282.

Research supported by the Australian Department of Science and Technology under the Queen Elizabeth II Fellowship Scheme, while this author was at the University of Newcastle, Australia.

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Bender, E.A., Richmond, L.B., Robinson, R.W. et al. The asymptotic number of acyclic digraphs. I. Combinatorica 6, 15–22 (1986). https://doi.org/10.1007/BF02579404

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AMS (1980) subject classification

  • 05 C 30
  • 05 C 20