Consider a graph on seven vertices. For each pair of vertices, you draw an edge between that pair of vertices with probability . Let be the probability that graph is connected. Then
where are relatively prime positive integers. Submit .
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Using a computer program, I found the number of graphs on seven vertices that have e edges and k connected components: e \ k 0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 2 0 2 1 ∑ 1 0 0 0 0 0 0 1 6 8 0 7 6 8 2 9 5 1 5 6 5 5 5 2 5 8 1 2 5 3 3 1 5 0 6 3 4 3 1 4 0 2 9 0 7 4 5 2 0 2 7 5 5 1 1 6 1 7 5 5 4 2 5 7 2 0 3 4 9 5 9 8 5 1 3 3 0 2 1 0 2 1 1 1 8 6 6 2 5 6 2 0 0 0 0 0 1 3 3 7 7 3 2 4 1 7 4 5 3 6 0 4 5 9 9 0 3 5 5 9 5 2 1 1 8 9 9 5 7 6 3 1 8 5 7 3 5 1 0 5 7 0 0 0 0 0 0 2 0 7 5 3 6 3 0 0 0 0 5 2 5 0 6 7 6 2 5 0 0 5 2 6 2 5 9 4 5 2 1 0 2 1 0 0 0 0 0 0 0 0 0 0 0 2 0 8 1 8 4 0 0 0 1 2 9 5 7 3 5 2 1 0 3 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 2 7 5 5 0 0 2 1 0 3 5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 4 5 6 0 2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 7 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Then Q ( p ) = 1 6 8 0 7 p 6 ( 1 − p ) 1 5 + 6 8 2 9 5 p 7 ( 1 − p ) 1 4 + 1 5 6 5 5 5 p 8 ( 1 − p ) 1 3 + 2 5 8 1 2 5 p 9 ( 1 − p ) 1 2 + 3 3 1 5 0 6 p 1 0 ( 1 − p ) 1 1 + 3 4 3 1 4 0 p 1 1 ( 1 − p ) 1 0 + 2 9 0 7 4 5 p 1 2 ( 1 − p ) 9 + 2 0 2 7 5 5 p 1 3 ( 1 − p ) 8 + 1 1 6 1 7 5 p 1 4 ( 1 − p ) 7 + 5 4 2 5 7 p 1 5 ( 1 − p ) 6 + 2 0 3 4 9 p 1 6 ( 1 − p ) 5 + 5 9 8 5 p 1 7 ( 1 − p ) 4 + 1 3 3 0 p 1 8 ( 1 − p ) 3 + 2 1 0 p 1 9 ( 1 − p ) 2 + 2 1 p 2 0 ( 1 − p ) + p 2 1 . Hence, ∫ 0 1 Q ( p ) d p = 3 6 9 5 1 2 2 4 2 5 6 5 .