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

Evaluating EYM amplitudes in four dimensions by refined graphic expansion