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Abstract for Richard A. Brualdi and Stephen Mellendorf, Two Extremal Problems in Graph Theory 

We consider the following two problems.
(1) Let $t$ and $n$ be positive integers with $n\geq t\geq 2$. Determine
the maximum number of edges of a graph \break
of order  $n$ that contains neither $K_t$ nor $K_{t,t}$ as a subgraph.
(2) Let $r$,  $t$ and $n$ be positive integers with $n\geq rt$ and $t\geq 2$.

 Determine the maximum number of edges of a graph 
of order  $n$ that does not contain $r$  disjoint copies of $K_t$.
Problem 1 for $n<2t$ is solved by Tur\'{a}n's theorem and we 
solve it  for $n=2t$. We also solve  Problem 2 for $n=rt$.
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