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{\bf Ralucca Gera and Jian Shen}
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{\bf Extension of Strongly Regular Graphs}
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The Friendship Theorem states that if any two people in a party have
exactly one common friend, then there exists a politician who is a
friend of everybody. In this paper, we generalize the Friendship
Theorem. Let $\lambda$ be any nonnegative integer and $\mu$ be any
positive integer. Suppose each pair of friends have exactly
$\lambda$ common friends and each pair of strangers have exactly
$\mu$ common friends in a party. The corresponding graph is a
generalization of strongly regular graphs obtained by relaxing the
regularity property on vertex degrees. We prove that either everyone
has exactly the same number of friends or there exists a politician
who is a friend of everybody. As an immediate consequence, this
implies a recent conjecture by Limaye et.~al.
\bye