connected = truefalse 

If the option connected is specified, the graph created is connected, and hence has at least n-1 edges.

For RandomGraph(n,m,connected), m must be at least n-1. A random tree is first created, then the remaining m-n+1 edges are

For RandomGraph(n,p,connected), a random tree is first created then each remaining edge is present with probability p.

degree = nonnegint

If the option degree=d is specified, and d-regular n vertex graph is possible, then a random d-regular graph having n vertices will be returned. Note that this option cannot be present with the directed option. This is equivalent to using the RandomRegularGraph command.

directed = truefalse 

If the option directed is specified, a random directed graph is chosen. This is equivalent to using the RandomDigraph command. Default value is false.

seed = integer or none

Seed for the random number generator. When an integer is specified, this is equivalent to calling randomize(seed).

weights = range

If the option weights=m..n is specified, where $m\le n$ are integers, the graph is a weighted graph with integer edge weights chosen from [m,n] uniformly at random.  The weight matrix W in the graph has datatype=integer, and if the edge from vertex i to j is not in the graph then W[i,j] = 0.

If the option weights=x..y where $x\le y$ are decimals is specified, the graph is a weighted graph with numerical edge weights chosen from [x,y] uniformly at random.  The weight matrix W in the graph has datatype=float[8], that is, double precision floats (16 decimal digits), and if the edge from vertex i to j is not in the graph then W[i,j] = 0.0.

If the option weights=f where f is a function (a Maple procedure) that returns a number (integer, rational, or decimal number), then f is used to generate the edge weights.  The weight matrix W in the graph has datatype=anything, and if the edge from vertex i to j is not in the graph then W[i,j] = 0.

RandomGraph(n,p) creates an undirected unweighted graph on n vertices where each possible edge is present with probability p where $0.0\le p\le 1.0$.

RandomGraph(n,m) creates an undirected unweighted graph on n vertices and m edges where the m edges are chosen uniformly at random. The value of m must satisfy $0\le m\le \mathrm{binomial}\left(n\,2\right)\=n\frac{n-1}{2}$.

If the first input is a positive integer n, then the vertices are labeled 1,2,...,n.  Alternatively, you may specify the vertex labels in a list.

This model of random graph generation, in which edges are selected with uniform probability from all possible edges in a graph on the specified vertices, is known as the Erdős–Rényi model.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{with}\left(\mathrm{RandomGraphs}\right)\:$

$G≔\mathrm{RandomGraph}\left(8\,0.5\right)$

$G≔\mathrm{Graph\; 1:\; an\; undirected\; graph\; with\; 8\; vertices\; and\; 10\; edges}$

$G≔\mathrm{RandomGraph}\left(8\,10\right)$

$G≔\mathrm{Graph\; 2:\; an\; undirected\; graph\; with\; 8\; vertices\; and\; 10\; edges}$

$G≔\mathrm{RandomGraph}\left(8\,10\,\mathrm{connected}\right)$

$G≔\mathrm{Graph\; 3:\; an\; undirected\; graph\; with\; 8\; vertices\; and\; 10\; edges}$

$\mathrm{IsConnected}\left(G\right)$

$\mathrm{true}$

$G≔\mathrm{RandomGraph}\left(6\,\mathrm{degree}=3\right)$

$G≔\mathrm{Graph\; 4:\; an\; undirected\; graph\; with\; 6\; vertices\; and\; 9\; edges}$

$\mathrm{IsRegular}\left(G\right)$

$\mathrm{true}$

$H≔\mathrm{RandomGraph}\left(4\,1.0\,\mathrm{weights}=0...1.0\right)$

$H≔\mathrm{Graph\; 5:\; an\; undirected\; weighted\; graph\; with\; 4\; vertices\; and\; 6\; edges}$

$\mathrm{WeightMatrix}\left(H\right)$

$\left[\begin{array}{cccc}0.& 0.809734551911930& 0.230156065952094& 0.761731208483085\\ 0.809734551911930& 0.& 0.158057578940872& 0.580956679189321\\ 0.230156065952094& 0.158057578940872& 0.& 0.423165119881119\\ 0.761731208483085& 0.580956679189321& 0.423165119881119& 0.\end{array}\right]$

$H≔\mathrm{RandomGraph}\left(8\,10\,\mathrm{connected}\,\mathrm{weights}=1..4\right)$

$H≔\mathrm{Graph\; 6:\; an\; undirected\; weighted\; graph\; with\; 8\; vertices\; and\; 10\; edges}$

$\mathrm{WeightMatrix}\left(H\right)$

$\left[\begin{array}{cccccccc}0& 4& 1& 0& 0& 2& 0& 0\\ 4& 0& 0& 1& 0& 0& 2& 0\\ 1& 0& 0& 3& 2& 0& 3& 0\\ 0& 1& 3& 0& 0& 0& 4& 4\\ 0& 0& 2& 0& 0& 0& 0& 0\\ 2& 0& 0& 0& 0& 0& 0& 0\\ 0& 2& 3& 4& 0& 0& 0& 0\\ 0& 0& 0& 4& 0& 0& 0& 0\end{array}\right]$

$U≔\mathrm{rand}\left(1..4\right)\:$

f := proc() local x; x := U(); if x=1 then 1 else 2 end if; end proc:

$H≔\mathrm{RandomGraph}\left(6\,1.0\,\mathrm{weights}=f\right)$

$H≔\mathrm{Graph\; 7:\; an\; undirected\; weighted\; graph\; with\; 6\; vertices\; and\; 15\; edges}$

$\mathrm{WeightMatrix}\left(H\right)$

$\left[\begin{array}{cccccc}0& 2& 1& 1& 2& 2\\ 2& 0& 1& 1& 2& 1\\ 1& 1& 0& 2& 1& 2\\ 1& 1& 2& 0& 2& 2\\ 2& 2& 1& 2& 0& 2\\ 2& 1& 2& 2& 2& 0\end{array}\right]$