We define an equivalence relation on graphs with signed edges, such that the associated adjacency matrices of two equivalent graphs are congruent over $\mathbb{ Z}$. We show that signed graphs whose eigenvalues are larger than $-2$ are equivalent to one of the simply laced Dynkin diagrams: $A_{ n}$, $D_{ n}$, $E_{ 6}$, $E_{ 7}$ and $E_{ 8}$. Checkerboard graph links are a class of fibred strongly quasipositive links which include positive braid links. We use the previous result to prove that a checkerboard graph link with maximal signature is isotopic to one of the links realized by the simply laced Dynkin diagrams.
Categories: math.GT