The spectrum of a graph is the set of eigenvalues of its adjacency matrix. A group, together with a multiset of elements of the group, gives a Cayley graph, and a semidirect product provides a method of producing new groups. This paper compares the spectra of cyclic groups to those of their semidirect products, when the products exist. It was found that many of the interesting identities that result can be described through number theory, field theory, and representation theory. The main result of this paper gives a formula that can be used to find the spectrum of semidirect products of cyclic groups.

Author Bio

Nathan Fox is an undergraduate at the University of Minnesota-Twin Cities, where he is majoring in mathematics and computer science. For eight weeks in the summer of 2010, he worked at the Canisius College REU program, Geometry and Physics on Graphs, in Buffalo, New York. After graduating in 2012, he plans to pursue a graduate degree in either theoretical math or computer science. Areas of interest include group theory, representation theory, graph theory, field theory, number theory, cryptology, and formal language theory. His hobbies include bicycling and reading about interesting problems in mathematics and theoretical physics.