Question

(7) Let V = {ui, U2 . . . . Un} with n > 4. In this exercise we will compute the probability that in a random graph with vertex set V we have that v and v2 have an edge between them or have an edge to a common vertex (i.e, have a common neighbour) (If you are troubled by my use of the term random we choose a graph on n vertices uniformly at random from the set of all graphs with vertex set V. Since there are finitely many graphs with the vertex set V, such uniform distribution exists.) Let Gn be the set of all graphs on n vertices, and let An be the set of graphs on n vertices in which vi and ½ have an edge between them or have a common neighbour (note that An includes those graphs in which v1 and v2 have an edge between them and also have a common neighbour). Thus for a fixed value of n, the probability that in a random graph on n vertices we have that ul and v2 have an edge between them or have an edge to a common vertex is (a) For how many graphs on n vertices is there an edge between v and v2? Justify your response (b) How many graphs on n vertices satisfy both of the following properties . v and v2 are not adjacent; and . vi and v2 have no common neighbour Hint Describe a bijection between such graphs and elements of 10, 1,21n-2 x Gn-2. You do not need to carefully verify that the proposed mapping is a bijection (c) Using part (b), explain why IAn-2(3)-1 + (2(2)-1-3n-22 (d) Evaluate ) TL (e) Interpret the limit from part (d) in the context of graphs with many vertices. L.e., If we take a random graph with many 1 million) vertices, what does (d) tell us about , 2 and their neighbours

Answer 1

(7) Let V = {ui, U2 . . . . Un} with n > 4. In this exercise we will compute the probability that in a random graph with vertex set V we have that v and v2 have an edge between them or have an edge to a common vertex (i.e, have a common neighbour) (If you are troubled by my use of the term random we choose a graph on n vertices uniformly at random from the set of all graphs with vertex set V. Since there are finitely many graphs with the vertex set V, such uniform distribution exists.) Let Gn be the set of all graphs on n vertices, and let An be the set of graphs on n vertices in which vi and ½ have an edge between them or have a common neighbour (note that An includes those graphs in which v1 and v2 have an edge between them and also have a common neighbour). Thus for a fixed value of n, the probability that in a random graph on n vertices we have that ul and v2 have an edge between them or have an edge to a common vertex is (a) For how many graphs on n vertices is there an edge between v and v2? Justify your response (b) How many graphs on n vertices satisfy both of the following properties . v and v2 are not adjacent; and . vi and v2 have no common neighbour Hint Describe a bijection between such graphs and elements of 10, 1,21n-2 x Gn-2. You do not need to carefully verify that the proposed mapping is a bijection (c) Using part (b), explain why IAn-2(3)-1 + (2(2)-1-3n-22 (d) Evaluate ) TL (e) Interpret the limit from part (d) in the context of graphs with many vertices. L.e., If we take a random graph with many 1 million) vertices, what does (d) tell us about , 2 and their neighbours

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