zbMATH — the first resource for mathematics

Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
On the evolution of random graphs. (English) Zbl 0103.16301
A random graph ${{\Gamma }}_{n,N}$ is a undirected finite graph without parallel edges and slings. ${{\Gamma }}_{n,N}$ has $n$ points ${P}_{1},···,{P}_{n}$ and $N$ edges (${P}_{i},{P}_{j}\right)$, which are chosen at random so that all $\left(\genfrac{}{}{0pt}{}{\left(\genfrac{}{}{0pt}{}{n}{2}\right)}{N}\right)={C}_{n,N}$ possible choices are supposed to be equiprobable. Let be ${P}_{n,N}\left(A\right)={A}_{n,N}/{C}_{n,N}$ the probability that ${{\Gamma }}_{n,N}$ has the property $A$, where ${A}_{n,N}$ denotes the number of graphs with the given points ${P}_{1},···,{P}_{n}$, with $N$ edges $\left({P}_{i},{P}_{j}\right)$ and with the property $A$. ${{\Gamma }}_{n,N}$ is studied under the condition that $N$ is increased, i.e. if $N$ is equal, or asymptotically equal, to a given function $N\left(n\right)$ of $n$. For many properties $A$ there is shown that there exists a ”threshold function” $A\left(n\right)$ of the property $A$ tendig monotonically to $+\infty$ for $n\to +\infty$ such that ${lim}_{n\to +\infty }{P}_{n,N\left(n\right)}\left(A\right)=0$ or =1 if ${lim}_{n\to +\infty }\frac{N\left(n\right)}{A\left(n\right)}=0$ or $=+\infty$. $A\left(n\right)$ is a ”regular threshold function” of $A$ if there exists a probability distribution function $F\left(x\right)$ such that ${lim}_{n\to +\infty }{P}_{n,N\left(n\right)}\left(A\right)=F\left(x\right)$ if ${lim}_{n\to +\infty }\frac{N\left(n\right)}{A\left(n\right)}=x$, where $0 and $x$ is a point of continuity of $F\left(x\right)$. The investigated properties are as follows: the presence of certain subgraphs (e. g. trees, complete subgraphs, cycles, etc.) or connectedness, number of components etc. The results are of the following type: Theorem 3a. Suppose that $N\left(n\right)\sim cn$, where $c>0$. Let ${\gamma }_{k}$ denote the number of cycles of order $k$ contained in ${{\Gamma }}_{n,N}$ $\left(k=3,4,···\right)$. Then we have ${lim}_{n\to +\infty }{P}_{n,N\left(n\right)}\left({\gamma }_{k}=j\right)={\lambda }^{j}{e}^{-\lambda }/j!$, where $j=0,1,···$ and $\lambda ={\left(2c\right)}^{k}/2k$. Thus the threshold distribution corresponding to the threshold function $A\left(n\right)=n$ for the property that the graph contains a cycle of order $k$ is $1-{e}^{-{\left(2c\right)}^{k}/2k}$.
Reviewer: K.Čulik

MSC:
 05C80 Random graphs
topology