The determinant of the adjacency matrix of a quaternion unit gain graph

In this paper, we present a combinatorial description of the determinant of the adjacency matrix of a quaternion unit gain graph using recently introduced row-column noncommutative determinants by one of the authors. We define a quaternion unit gain graph as a graph in which each edge's orientation is assigned a quaternion unit, and its opposite orientation is assigned the inverse of this quaternion unit. Initially, we provide detailed combinatorial descriptions of the determinants of the adjacency matrices for a single cycle and a path graph with quaternion unit gains. Subsequently, we investigate the determinant of the adjacency matrix for quaternion unit gain graphs whose underlying graphs consist of multiple cycles and/or path graphs. We introduce a decomposition procedure for such graphs involving reductions obtained by cutting off edges associated with branch vertices so that each reduction's adjacency matrix is equal to the direct sum of its components' adjacency matrices. Our resulting theorem offers a combinatorial description for obtaining the determinant of an adjacency matrix in terms of cycle and graph path adjacency determinants on which they are decomposed. The obtained results are novel for quaternion unit gain graphs and complex ones, and they could be applied to various types of gain graphs, not just those with unit gains.