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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
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.
Received May 5, 2016, published May 18, 2016
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
Linking options:
https://www.mathnet.ru/eng/semr682 https://www.mathnet.ru/eng/semr/v13/p375
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