Spectra of quaternion unit gain graphs

A quaternion unit gain graph is a graph where each orientation of an edge is given a quaternion unit, which is the inverse of the quaternion unit assigned to the opposite orientation. In this paper we define the adjacency, Laplacian and incidence matrices for a quaternion unit gain graph and study their properties. These properties generalize several fundamental results from spectral graph theory of ordinary graphs, signed graphs and complex unit gain graphs. Bounds for both the left and right eigenvalues of the adjacency and Laplacian matrix are developed, and the right eigenvalues for the cycle and path graphs are explicitly calculated.