Random Graphs

This quiz is designed to assess your understanding of Random Graphs, a branch of Graph Theory that deals with the study of graphs with random properties.

14 Questions Published

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Questions

Question 1 Multiple Choice (Single Answer)

What is the expected number of edges in a random graph with (n) vertices and (p) probability of an edge between any two vertices?

\(np\)
\(n(n-1)p\)
\(n(n-1)p/2\)
\(n(n+1)p/2\)

Question 2 Multiple Choice (Single Answer)

What is the probability that a random graph with (n) vertices and (p) probability of an edge between any two vertices is connected?

\((1-p)^{n-1}\)
\(1-(1-p)^{n-1}\)
\(p^{n-1}\)
\(1-p^{n-1}\)

Question 3 Multiple Choice (Single Answer)

What is the expected number of components in a random graph with (n) vertices and (p) probability of an edge between any two vertices?

\(n-1\)
\(n\)
\(n+1\)
\(n(n-1)p\)

Question 4 Multiple Choice (Single Answer)

What is the probability that a random graph with (n) vertices and (p) probability of an edge between any two vertices is a tree?

\(p^{n-2}\)
\(1-p^{n-2}\)
\((1-p)^{n-2}\)
\(1-(1-p)^{n-2}\)

Question 5 Multiple Choice (Single Answer)

What is the expected number of cycles of length (k) in a random graph with (n) vertices and (p) probability of an edge between any two vertices?

\(n^{k-1}p^{k}\)
\(n^{k}p^{k}\)
\(n^{k+1}p^{k}\)
\(n^{k-2}p^{k}\)

Question 6 Multiple Choice (Single Answer)

What is the probability that a random graph with (n) vertices and (p) probability of an edge between any two vertices is Eulerian?

\((1-p)^{n-1}\)
\(1-(1-p)^{n-1}\)
\(p^{n-1}\)
\(1-p^{n-1}\)

Question 7 Multiple Choice (Single Answer)

What is the expected number of Hamiltonian cycles in a random graph with (n) vertices and (p) probability of an edge between any two vertices?

\(n!p^{n}\)
\(n!p^{n-1}\)
\(n!p^{n-2}\)
\(n!p^{n+1}\)

Question 8 Multiple Choice (Single Answer)

What is the probability that a random graph with (n) vertices and (p) probability of an edge between any two vertices is planar?

\((1-p)^{n-3}\)
\(1-(1-p)^{n-3}\)
\(p^{n-3}\)
\(1-p^{n-3}\)

Question 9 Multiple Choice (Single Answer)

What is the expected number of matchings in a random graph with (n) vertices and (p) probability of an edge between any two vertices?

\(n!p^{n/2}\)
\(n!p^{n-1}\)
\(n!p^{n-2}\)
\(n!p^{n+1}\)

Question 10 Multiple Choice (Single Answer)

What is the probability that a random graph with (n) vertices and (p) probability of an edge between any two vertices is perfect?

\((1-p)^{n-1}\)
\(1-(1-p)^{n-1}\)
\(p^{n-1}\)
\(1-p^{n-1}\)

Question 11 Multiple Choice (Single Answer)

What is the expected number of cliques of size (k) in a random graph with (n) vertices and (p) probability of an edge between any two vertices?

\(n^{k}p^{k}\)
\(n^{k-1}p^{k}\)
\(n^{k+1}p^{k}\)
\(n^{k-2}p^{k}\)

Question 12 Multiple Choice (Single Answer)

What is the probability that a random graph with (n) vertices and (p) probability of an edge between any two vertices is strongly connected?

\((1-p)^{n-1}\)
\(1-(1-p)^{n-1}\)
\(p^{n-1}\)
\(1-p^{n-1}\)

Question 13 Multiple Choice (Single Answer)

What is the expected number of independent sets of size (k) in a random graph with (n) vertices and (p) probability of an edge between any two vertices?

\(n^{k}p^{k}\)
\(n^{k-1}p^{k}\)
\(n^{k+1}p^{k}\)
\(n^{k-2}p^{k}\)

Question 14 Multiple Choice (Single Answer)

What is the probability that a random graph with (n) vertices and (p) probability of an edge between any two vertices is Hamiltonian?

\((1-p)^{n-1}\)
\(1-(1-p)^{n-1}\)
\(p^{n-1}\)
\(1-p^{n-1}\)

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