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Hello,

I have a problem in combinatorics and graph theory I'm struggling with.

$G(n,m)$ is the family of all graphs with $n$ vertices and $m$ edges (I consider $m < n$). Each graph in $G(n,m)$ is selected with uniform probability. What is the probability that the graph selected has exactly $c$ connected components?

An equivalent question is: what is the probability that exactly $k$ edges should be removed from the selected graph in order to make it a forest (graph without cycles)?

There is a solution for the case of $k = 0$ here.

Thank you.

1

# Number of connected components in a graph from G(n,m)

Hello,

I have a problem in combinatorics and graph theory I'm struggling with.

$G(n,m)$ is the family of all graphs with $n$ vertices and $m$ edges (I consider $m < n$). Each graph in $G(n,m)$ is selected with uniform probability. What is the probability that the graph selected has exactly $c$ connected components?

An equivalent question is: what is the probability that exactly $k$ edges should be removed from the selected graph in order to make it a forest (graph without cycles)?

There is a solution for the case of $k = 0$ here.

Thank you.