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Asymptotic approximation for the number of n-vertex graphs of given diameter

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Abstract

We prove that, for fixed k ≥ 3, the following classes of labeled n-vertex graphs are asymptotically equicardinal: graphs of diameter k, connected graphs of diameter at least k, and (not necessarily connected) graphs with a shortest path of length at least k. An asymptotically exact approximation of the number of such n-vertex graphs is obtained, and an explicit error estimate in the approximation is found. Thus, the estimates are improved for the asymptotic approximation of the number of n-vertex graphs of fixed diameter k earlier obtained by Füredi and Kim. It is shown that almost all graphs of diameter k have a unique pair of diametrical vertices but almost all graphs of diameter 2 have more than one pair of such vertices.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    T. I. Fedoryaeva

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    T. I. Fedoryaeva

Authors
  1. T. I. Fedoryaeva
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Correspondence to T. I. Fedoryaeva.

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Original Russian Text © T.I. Fedoryaeva, 2017, published in Diskretnyi Analiz i Issledovanie Operatsii, 2017, Vol. 24, No. 2, pp. 68–86.

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Fedoryaeva, T.I. Asymptotic approximation for the number of n-vertex graphs of given diameter. J. Appl. Ind. Math. 11, 204–214 (2017). https://doi.org/10.1134/S1990478917020065

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  • DOI: https://doi.org/10.1134/S1990478917020065

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