This is a follow-up to this question.

For any set $X$ we set $[X]^2 = \big\{\{x,y\}: x\neq y \in X\big\}$. If $G=(V,E)$ is a simple undirected graph, and $v\in V$, we set $N(v) = \{w\in V:\{v,w\}\in E\}$.

Let $\kappa$ be an infinite cardinal and suppose $c: [\kappa]^2 \to \kappa$ is an arbitrary function.

Is there $E \subseteq [\kappa]^2$ such that the graph $(\kappa, E)$ has the following property?

For all $\{a,b\}\in [\kappa]^2$ we have $c(\{a,b\}) = |N(a)\cap N(b)|$.