  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.
