Fomichev–Karev conjecture on graph invariants

Let $\mathbf{G}$ be the set of isomorphism classes of finite simple graphs. For $G\in\mathbf{G}$, let $\varphi(G)$ be the graph invariant defined by

$$\varphi(G)=2^{-3|V(G)|}\sum_{E'\subset E(G)}(-2)^{|E'|}\chi_3(G|_{E'}),$$

where $\chi_3(G)$ is the number of proper vertex colorings of $G$ with three colors. Let

$$\psi(G)=\frac{1}{2^{2|V(G)|}}\sum_{U\subset V(G)}\left(-\frac{1}{2}\right)^{|V(G)|-|U|}2^{\operatorname{corank}(A(G|_U))},$$

where $G|_U$ is the induced subgraph on $U$, $A(G|_U)$ is its adjacency matrix over the field with two elements, and $\operatorname{corank}$ denotes matrix corank. Fomichev–Karev conjecture. For every $G\in\mathbf{G}$,

$$\varphi(G)=\psi(G).$$

Fomichev and Karev introduced these invariants in connection with graph Chmutov–Varchenko relations and the $\mathfrak{sl}(2)$-weight system. The conjecture is proved in the paper, so it is no longer open.