It is proved that there is spectral sequence converging to $H^*(M^G,R)$ with the E1 page given by the graph cohomology complex $C_A(G)$ where $A:=H^*(M,R)$.
My question is a sort of general one: when $A=H^*(S^1,\mathbb Z)=\mathbb Z [x]/\langle x^2\rangle$ where $x$ has degree 1 (or do we need $\mathbb Q$ for formality) then $C_A(G)$ should be computing the graph configuration space of the circle (because the higher massey products are 0).
But, at the same time in the paper https://arxiv.org/pdf/math/0412264.pdf, this is the algebra used to compute the chromatic graph cohomology whose graded euler characteristic is the chromatic polynomial. Why exactly should these be related? or did I make a mistake in my interpretation of the results?