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Sibirskie Èlektronnye Matematicheskie Izvestiya [Siberian Electronic Mathematical Reports], 2016, Volume 13, Pages 375–387
DOI: https://doi.org/10.17377/semi.2016.13.033
(Mi semr682)
 

This article is cited in 5 scientific papers (total in 5 papers)

Discrete mathematics and mathematical cybernetics

Structure of the diversity vector of balls of a typical graph with given diameter

T. I. Fedoryaeva

Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
Full-text PDF (217 kB) Citations (5)
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Abstract: For labeled $n$-vertex graphs with fixed diameter $d\geq 1$, the diversity vectors of balls (the ith component of the vector is equal to the number of different balls of radius $i$) are studied asymptotically. An explicit description of the diversity vector of balls of a typical graph with given diameter is obtained. A set of integer vectors $\Lambda_{n,d}$ consisting of $\lfloor\frac{d-1}{2}\rfloor$ different vectors for $d\geq 5$ and a unique vector for $d<5$ is found. It is proved that almost all labeled $n$-vertex graphs of diameter $d$ have the diversity vector of balls belonging to $\Lambda_ {n,d}$. It is established that this property is not valid after removing any vector from $\Lambda_ {n,d}$. A number of properties of a typical graph of diameter $d$ is proved. In particular, it is obtained that such a graph for $d\geq 3$ does not possess the local $2$-diversity of balls and at the same time has the local $1$-diversity of balls, but has the full diversity of balls if $d=1,2$.
Keywords: graph, labeled graph, distance, metric ball, number of balls, diversity vector of balls, typical graph.
Funding agency Grant number
Russian Foundation for Basic Research 14-01-00507_а
Received May 5, 2016, published May 18, 2016
Document Type: Article
UDC: 519.1+519.173
MSC: 05C12
Language: Russian
Citation: T. I. Fedoryaeva, “Structure of the diversity vector of balls of a typical graph with given diameter”, Sib. Èlektron. Mat. Izv., 13 (2016), 375–387
Citation in format AMSBIB
\Bibitem{Fed16}
\by T.~I.~Fedoryaeva
\paper Structure of the diversity vector of balls of a typical graph with given diameter
\jour Sib. \`Elektron. Mat. Izv.
\yr 2016
\vol 13
\pages 375--387
\mathnet{http://mi.mathnet.ru/semr682}
\crossref{https://doi.org/10.17377/semi.2016.13.033}
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  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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