The expected number of edges in a random graph is given by the formula (E(X) = np), where (n) is the number of vertices and (p) is the probability of an edge between any two vertices.

The probability that a random graph is connected is given by the formula (1-(1-p)^{n-1}), where (n) is the number of vertices and (p) is the probability of an edge between any two vertices.

The expected number of components in a random graph is given by the formula (n), where (n) is the number of vertices.

The probability that a random graph is a tree is given by the formula (1-p^{n-2}), where (n) is the number of vertices and (p) is the probability of an edge between any two vertices.

The expected number of cycles of length (k) in a random graph is given by the formula (n^{k}p^{k}), where (n) is the number of vertices, (p) is the probability of an edge between any two vertices, and (k) is the length of the cycle.

The probability that a random graph is Eulerian is given by the formula (1-(1-p)^{n}), where (n) is the number of vertices and (p) is the probability of an edge between any two vertices.

The expected number of Hamiltonian cycles in a random graph is given by the formula (n!p^{n-1}), where (n) is the number of vertices and (p) is the probability of an edge between any two vertices.

The probability that a random graph is planar is given by the formula (1-(1-p)^{n-3}), where (n) is the number of vertices and (p) is the probability of an edge between any two vertices.

The expected number of matchings in a random graph is given by the formula (n!p^{n/2}), where (n) is the number of vertices and (p) is the probability of an edge between any two vertices.

The probability that a random graph is perfect is given by the formula (1-(1-p)^{n(n-1)/2}), where (n) is the number of vertices and (p) is the probability of an edge between any two vertices.

The expected number of cliques of size (k) in a random graph is given by the formula (n^{k}p^{k}), where (n) is the number of vertices, (p) is the probability of an edge between any two vertices, and (k) is the size of the clique.

The probability that a random graph is strongly connected is given by the formula (1-(1-p)^{n(n-1)/2}), where (n) is the number of vertices and (p) is the probability of an edge between any two vertices.

The expected number of independent sets of size (k) in a random graph is given by the formula (n^{k}(1-p)^{n-k}), where (n) is the number of vertices, (p) is the probability of an edge between any two vertices, and (k) is the size of the independent set.

The probability that a random graph is Hamiltonian is given by the formula (1-(1-p)^{n(n-1)/2}), where (n) is the number of vertices and (p) is the probability of an edge between any two vertices.