Эта публикация цитируется в
5 статьях
Дискретная математика и математическая кибернетика
Строение вектора разнообразия шаров типичного графа заданного диаметра
Т. И. Федоряева Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
Аннотация:
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
$d=1,2$.
Ключевые слова:
graph, labeled graph, distance, metric ball, number of balls, diversity vector of balls, typical graph.
УДК:
519.1+
519.173
MSC: 05C12 Поступила 5 мая 2016 г., опубликована
18 мая 2016 г.
DOI:
10.17377/semi.2016.13.033