Spectral Properties of Quaternionic Unit Gain Cycles

Abstract

The quaternions are a non-commutative division ring that extends the complex numbers. A gain graph is a simple graph together with a gain function that assigns a value from an arbitrary group to each edge of the graph. We can define certain concepts on these graphs such as adjacency and Laplacian matrices, gains of paths, and more. If we restrict ourselves to the unit norm quaternions, we can define quaternionic unit gain graphs, or U(H)-gain graphs, as gain graphs where the domain of the gain function is the unit quaternions. Traditional methods from spectral graph theory are not directly extended to quaternionic unit gain graphs due to non-commutativity. In this thesis, we extend a previous result from complex unit gain graphs, so that the right eigenvalues of the adjacency matrix for a U(H)-gain cycle can be written explicitly from the gain of the cycle. A thorough treatment of quaternions, quaternionic linear algebra, T-gain graphs, and U(H)-gain cycles is given. At the end, the right eigenvalues are calculated for a particular U(H)-gain cycle, and the results are compared to those obtained from a MATLAB method for approximating them.