Let G (V, E) be a loop free undirected graph (a) For each such graph, where V 3, find P(G, ) and show that in it the terms contain consecutive powers of Also show that the coefficients of these consecutive powers alternate in sign (b) Now consider G (V, E), where V n 4 and E k Prove by mathematical induction that the terms in P(G, ) contain consecutive powers of X and that the coefficients of these consecutive powers alternate in sign For the induction hypothesis, assume that the result is true for all loop free undirected graphs G (V, E), where either (i) V n 1 or (ii) V n, but E k 1 (c) Prove that if V n, then the coefficient of n 1in P(G, ) is the negative of E

Question: Let G = (V, E) be a loop-free undirected graph. (a) For each such graph, where |V| 3, find P(G, ) and show that

Let G = (V, E) be a loop-free undirected graph.
(a) For each such graph, where |V| ≤ 3, find P(G, λ) and show that in it the terms contain consecutive powers of λ. Also show that the coefficients of these consecutive powers alternate in sign.
(b) Now consider G = (V, E), where |V| = n ≥ 4 and |E| = k. Prove by mathematical induction that the terms in P(G, λ) contain consecutive powers of X and that the coefficients of these consecutive powers alternate in sign. [For the induction hypothesis, assume that the result is true for all loop-free undirected graphs G = (V, E), where either (i) |V| = n - 1 or (ii) |V| = n, but |E| = k - 1.]
(c) Prove that if |V| = n, then the coefficient of λn-1in P(G, λ) is the negative of |E|.

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