The recursive expansion of tree level multitrace Einstein-Yang-Mills (EYM) amplitudes induces a refined graphic expansion, by which any tree-level EYM amplitude can be expressed as a summation over all possible refined graphs. Each graph contributes a unique coefficient as well as a proper combination of color-ordered Yang-Mills (YM) amplitudes. This expansion allows one to evaluate EYM amplitudes through YM amplitudes, the latter have much simpler structures in four dimensions than the former. In this paper, we classify the refined graphs for the expansion of EYM amplitudes into N ^k MHV sectors. Amplitudes in four dimensions, which involve k + 2 negative-helicity particles, at most get non-vanishing contribution from graphs in N ^{k′ (k′ ≤ k)} MHV sectors. By the help of this classification, we evaluate the non-vanishing amplitudes with two negative-helicity particles in four dimensions. We establish a correspondence between the refined graphs for single-trace amplitudes with \( \left({g}_i^{-},{g}_j^{-}\right) \) or \( \left({h}_i^{-},{g}_j^{-}\right) \) configuration and the spanning forests of the known Hodges determinant form. Inspired by this correspondence, we further propose a symmetric formula of double-trace amplitudes with \( \left({g}_i^{-},{g}_j^{-}\right) \) configuration. By analyzing the cancellation between refined graphs in four dimensions, we prove that any other tree amplitude with two negative-helicity particles has to vanish.

树级多迹线Einstein-Yang-Mills（EYM）幅度的递归扩展引起精化图形扩展，通过该扩展，任何树级EYM幅度都可以表示为所有可能精化图的总和。每个图都贡献一个唯一的系数，以及颜色顺序的Yang-Mills（YM）振幅的适当组合。这种扩展允许通过YM振幅来评估EYM振幅，后者在四个维度上比前者更简单。在本文中，我们将EYM振幅扩展到N ^k MHV扇区的精简图分类。涉及k + 2个负螺旋粒子的四个维度上的振幅最多得到N ^{k′（k′≤ ķ）} MHV扇区。通过这种分类，我们可以在四个维度上评估两个负螺旋形粒子的不消失振幅。我们用\（\ left（{g} _i ^ {-}，{g} _j ^ {-} \ right）\）或\（\ left（{h} _i ^ {-}，{g} _j ^ {-} \ right）\）配置和已知Hodges行列式形式的生成林。受到这种对应关系的启发，我们进一步提出了具有\（\ left（{g} _i ^ {-}，{g} _j ^ {-} \ right）\）配置的双迹线振幅的对称公式。通过分析四个维度上精炼图之间的抵消，我们证明带有两个负螺旋性粒子的任何其他树幅都必须消失。