Consider a graph $G(V,E)$ where every vertice $v\in V$ has its weight $w(v)$. I want to work with $$\sum_{(u,v)\in E}w(u)w(v)$$ I think there should many good ways to approach this object, but what are they?

Furhermore, is there method for calculating this sum via incidence matrix?

An incidence matrix is a $|V|\times|V|$ $(0,1)$-matrix with entries $a(u,v)$, such that $$ a(u,v)= \begin{cases} 1 & (u,v)\in E \\ 0 & otherwise \end{cases} $$

In more details: $V=\{0,1\}^m$, a pair $(u,v)\in E$ iff $u+v$ is a Fibonacci tiling. I want to work with the "total edge weight". Is there any representations of it in terms of matrices and vectors?