Graph-based Bern-Carasso-Johansson (BCJ) relation for Berends-Giele currents in bi-adjoint scalar (BS) theory, which is characterized by connected tree graphs, was proposed in an earlier work. In this note, we provide a systematic study of this relation. We first prove the relations based on two special types of graphs: simple chains and star graphs. The general graph-based BCJ relation established by an arbitrary tree graph is further proved, through Berends-Giele recursion. When combined with proper off-shell extended numerators, this relation induces the graph-based BCJ relation for Berends-Giele currents in Yang-Mills theory. The corresponding relations for amplitudes are obtained via on-shell limits.

在早期的工作中提出了双伴随标量 (BS) 理论中基于图的 Bern-Carasso-Johansson (BCJ) 关系，其特征是连接的树图。在本说明中，我们对这种关系进行了系统研究。我们首先证明基于两种特殊类型图的关系：简单链和星图。通过Berends-Giele递归，进一步证明了由任意树图建立的一般的基于图的BCJ关系。当与适当的离壳扩展分子结合时，该关系在 Yang-Mills 理论中导出 Berends-Giele 电流的基于图的 BCJ 关系。振幅的对应关系是通过壳上极限获得的。