Spectral graph theory examines the structure of a graph by studying the eigenvalues of certain matrices associated with the graph. A Cayley graph is a construction that embeds the structure of a group generated by a certain generating set. Since graphs are a means to study groups, and linear algebra gives the spectral theorems to study graphs, the next logical step is to use spectral theory to examine finite groups.
Each vertex of the Cayley graph of a group corresponds to an element of the group. Choose a generating set for the group and right-multiply each group element by each element of the generating set. Draw the directed edge between the corresponding vertices in the graph, to .
The graphs shown here have differently colored edges, one color per generator.
Now take eigenvalues of the adjacency matrix of the graph and plot them in the complex plane.
If the generating set is symmetric, then the adjacency matrix is symmetric and has real eigenvalues, so nonsymmetric generating sets were chosen. Viewing the complex eigenvalues of the directed graph makes the patterns a bit easier to see.
You may notice that in the case of a cyclic group of order , the eigenvalues are precisely the roots of unity in the complex plane, which form a group isomorphic to the original cyclic group in question. A possible direction for future research would be to see if similar isomorphisms exist for other groups.