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Thesis - Campus Access Only
Master of Science (MS)
In graph theory, the Friendship Theorem states that any finite graph in which every two vertices share exactly one common neighbor has a vertex adjacent to all other vertices. We present a proof of this theorem by first considering a class of regular graphs called strongly regular graphs, and proving certain conditions on these graphs by using spectral properties of their adjacency matrices. We also present a second proof that involves counting walks and using modular arithmetic. Lastly, we look at infinite “ friendship graphs ” and show that the Friendship Theorem does not hold for infinite graphs. We then present a method for constructing infinite friendship graphs, look at some properties of those graphs, and pose some open questions about them.
Zimmermann, David Sawyer, "The Friendship Theorem" (2010). Master's Theses. 3798.