Prove that U = T 2 has an F distribution with 1 numerator and ν denominator degrees of freedom if T has a t distribution with ν df.

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T1.4: Let ẞ(G) be the minimum size of a vertex cover, a(G) be the maximum size of an independent set and m(G) = |E(G)|. (i) Prove that if G is triangle free (no induced K3) then m(G) ≤ a(G)B(G). Hints - The neighborhood of a vertex in a triangle free graph must be independent; all edges have at least one end in a vertex cover. (ii) Show that all graphs of order n ≥ 3 and size m> [n2/4] contain a triangle. Hints - you may need to use either elementary calculus or the arithmetic-geometric mean inequality.

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Chapter 7.2, Problem 33E

To determine

Prove that U=T2 has an F distribution with 1 numerator and ν denominator degrees of freedom if T has a t distribution with ν df.

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