Definition 4.17 (Signing). $S:V\times V\to ℝ$ is a signing of $G$ if

$$S(x,y)=\{\begin{array}{cc}\pm 1\hfill & \text{if }A(x,y)=1\hfill \\ 0\hfill & \text{if }A(x,y)=0\hfill \end{array}$$

and $S(x,y)=S(y,x)$ (symmetric). So can think of $S$ as a function $E\to \{\pm 1\}$.