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ЖУРНАЛЫ // Сибирские электронные математические известия // Архив

Сиб. электрон. матем. изв., 2016, том 13, страницы 375–387 (Mi semr682)

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



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