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Эта публикация цитируется в 1 научной статье (всего в 1 статье)
Дискретная математика и математическая кибернетика
Logarithmic asymptotics of the number of central vertices of almost all $n$-vertex graphs of diameter $k$
T. I. Fedoryaeva Sobolev Institute of Mathematics, pr. Koptyuga, 4, 630090, Novosibirsk, Russia
Аннотация:
The asymptotic behavior of the number of central vertices and F. Buckley's central ratio ${\mathbb R}_{c}(G)=|{\mathbb C}(G)|/|V(G)|$ for almost all $n$-vertex graphs $G$ of fixed diameter $k$ is investigated.
The logarithmic asymptotics of the number of central vertices for almost all such $n$-vertex graphs is established: $0$ or $\log_2 n$ ($1$ or $\log_2 n$), respectively, for arising here subclasses of graphs of the even (odd) diameter.
It is proved that for almost all $n$-vertex graphs of diameter $k$, ${\mathbb R}_{c}(G)=1$ for $k=1,2$, and ${\mathbb R}_{c }(G)=1-2/n$ for graphs of diameter $k=3$, while for $k\geq 4$ the value of the central ratio ${\mathbb R}_{c}(G)$ is bounded by the interval $(\frac{\Delta}{6} + r_1(n), 1-\frac{\Delta}{6} - r_1(n))$ except no more than one value (two values) outside the interval for even diameter $k$ (for odd diameter $k$) depending on $k$. Here $\Delta\in (0,1)$ is arbitrary predetermined constant and $r_1(n),r_2(n)$ are positive infinitesimal functions.
Ключевые слова:
graph, diameter, radius, central vertices, number of central vertices, central ratio, center, spectrum of center, typical graphs, almost all graphs.
Поступила 11 мая 2022 г., опубликована 11 ноября 2022 г.
Образец цитирования:
T. I. Fedoryaeva, “Logarithmic asymptotics of the number of central vertices of almost all $n$-vertex graphs of diameter $k$”, Сиб. электрон. матем. изв., 19:2 (2022), 747–761
Образцы ссылок на эту страницу:
https://www.mathnet.ru/rus/semr1536 https://www.mathnet.ru/rus/semr/v19/i2/p747
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