Many objects of interest in algebraic topology can be computed by graph complexes. This includes homotopy groups of embedding spaces and diffeomorphism groups, and in particular (parts of) the cohomology of the moduli space of curves. Unfortunately, the graph homology itself is still a mysterious object and far from fully understood. In my talk, I will introduce a smaller quasi-isomorphic variant of the most basic graph complex (the commutative graph complex of Kontsevich), and discuss the present state of knowledge of the graph homology. The talk is based on arXiv:2503.17131 and arXiv:2508.13724.