In this paper, we study bearing equivalence in directed graphs. We first give a strengthened definition of bearing equivalence based on the \textit{kernel equivalence} relationship between bearing rigidity matrix and bearing Laplacian matrix. We then present several conditions to characterize bearing equivalence for both directed acyclic and cyclic graphs. These conditions involve the spectrum and null space of the associated bearing Laplacian matrix for a directed bearing formation. For directed acyclic graphs, all eigenvalues of the associated bearing Laplacian are real and nonnegative, while for directed graphs containing cycles, the bearing Laplacian can have eigenvalues with negative real parts. Several examples of bearing equivalent and bearing non-equivalent formations are given to illustrate these conditions.

翻译：本文研究了有向图中的方位等价性。我们首先基于方位刚度矩阵与方位拉普拉斯矩阵之间的核等价关系，给出方位等价性的强化定义。随后提出若干条件来刻画有向无环图和有向环图中的方位等价性。这些条件涉及有向方位编队中相关方位拉普拉斯矩阵的谱与零空间。对于有向无环图，相关方位拉普拉斯矩阵的所有特征值均为实数且非负；而对于含有环的有向图，方位拉普拉斯矩阵可能存在具有负实部的特征值。本文通过给出若干方位等价与非等价编队实例来阐释这些条件。