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English version:
Journal of Applied and Industrial Mathematics, 2017, 11:2, 204-214

Volume 24, No 2, 2017, P. 68-86

UDC 519.1+519.175
T. I. Fedoryaeva
Asymptotic approximation for the number of n-vertex graphs of given diameter

Abstract:
We prove that, for fixed $k \ge 3$, 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.
Illustr. 3, bibliogr. 9.

Keywords: graph, labeled graph, shortest path, graph diameter, number of graphs, ordinary graph.

DOI: 10.17377/daio.2017.24.534

Tatiana I. Fedoryaeva 1,2
1. Sobolev Institute of Mathematics,
4 Koptyug Ave., 630090 Novosibirsk, Russia
2. Novosibirsk State University,
2 Pirogov St., 630090 Novosibirsk, Russia
e-mail: tatiana.fedoryaeva@gmail.com

Received 29 March 2016
Revised 4 July 2016

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