Combinatorics Seminar: Spectral Faux Tees and Gluing Sets

Abstract: A classic problem in mathematics is "Can you hear the shape of a drum?" In graph theory, this is often phrased in terms of the eigenvalues of matrices associated with a graph, and whether a particular set of eigenvalues is associated with a unique graph. We will look at the existence and construction of graphs which are not trees but share eigenvalues with some tree (these are the "spectral faux trees" and examples where we "cannot hear the shape of a tree"). The answer to this problem varies significantly with which type of matrix is considered and we will consider several common matrices showing the different strengths and weaknesses of matrices. We will also present a result that unexpectedly came up during the investigation of spectral faux trees that deals with the results of gluing graphs.

Joint work with Elena D'Avanzo, Rachel Heikkinen, Joel Jeffries, Alyssa Kruczek, and Harper Niergarth