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.

中文翻译：

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四元数单位增益图谱

四元数单位增益图是其中一条边的每个方向都被赋予一个四元数单位的图，该四元数单位是分配给相反方向的四元数单位的倒数。在本文中，我们定义了四元数单位增益图的邻接矩阵、拉普拉斯矩阵和关联矩阵，并研究了它们的性质。这些性质概括了普通图、有符号图和复单位增益图的谱图理论的几个基本结果。开发了邻接矩阵和拉普拉斯矩阵的左右特征值的边界，并明确计算了循环图和路径图的右特征值。