Past exam of the graph Theory and Probabilistic Combinatorics course of the University of Oxford / 2009 / A / 2 / c

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- Solution c

Solution

t = - x > 0

. The coefficient formula gives

(- 1)^{n} p_{G} (x) = \sum_{i = 0}^{n - 1} a_{i} t^{n - i} .

Every summand is nonnegative and the leading term is

t^{n} > 0

, since

a_{0} = 1

. Therefore

(- 1)^{n} p_{G} (x) > 0 (x < 0) .

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