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Date of Award
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. Paper 3798.