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AMS-MAA Joint Invited Address

Eigenvalues and Graphs

Thursday, July 29, 10:00 a.m. - 10:50 a.m., Marriott Philadelphia Downtown, Grand Ballroom Salon G & H

Steven Butler, Iowa State University

One way to store information about a graph is by an array with entries indexed by pairs of vertices with each entry giving information about a relationship between the pair. The linear algebraist in us would say, ``let's change our names and instead of calling it an array, let us call it a matrix, which is an array with benefits''. Among these benefits are the eigenvalues and singular values of the matrix. The eigenvalues give information about the linear transformation to which the matrix corresponds, and this can capture some structural properties of the graph (often with just knowing a few of the extremal eigenvalues). This provides a way to obtain information about a graph with just a handful of parameters. We will explore several different possible matrices and look at some of the information that we can, and in some cases cannot, learn by studying the eigenvalues.

Steve Butler is the Barbara J Janson Professor of Mathematics at Iowa State University. He earned his PhD degree in 2011 from UC San Diego where he studied spectral graph theory under Fan Chung. He has worked extensively with Ron Graham, and is (currently) the last person to get an Erdos number of one. He has published over 70 papers in mathematics in topics ranging from circle packings and permutation enumeration to origami and card shuffling; has performed at the Iowa State Fair; and is co-author on the forthcoming book "Juggling Counts". More information about his research and teaching is at