I have a question about the combinatorial Laplacian $\Delta$ which is defined by $$\Delta(u,v)=c(u)1_{u=v}-c(u,v)$$ where $u, v$ are some vertices in the graph $G=(V, E)$, and $c(u,v)$ is a conductance function defined on the edge $uv$ (i.e. weighted functions).

If I define a function $F: V\to \mathbb{R}$, we can define the gradient $\nabla F(e)$ by $$\nabla F(uv):=c(u,v)(F(v)-F(u))$$. But how to understand the $\Delta F(uv)$ by the combinatorial Laplacian $\Delta$? Actually, textbook claims that $$\nabla \cdot \nabla F= -\Delta F$$

I have no idea to prove that.

I have a question about the combinatorial Laplacian $\Delta$ which is defined by $$\Delta(u,v)=c(u)1_{u=v}-c(u,v)$$ where $u, v$ are some vertices in the graph $G=(V, E)$, and $c(u,v)$ is a conductance function defined on the edge $uv$ (i.e. weighted functions).

If I define a function $F: V\to \mathbb{R}$, we can define the gradient $\nabla F(e)$ by $$\nabla F(uv):=c(u,v)(F(v)-F(u))$$. But how to understand the $\Delta F(uv)$ by the combinatorial Laplacian $\Delta$? Actually, textbook claims that $$\nabla \cdot \nabla F= -\Delta F$$

I have no idea to prove that.

The divergence $\nabla\cdot f$ is defined by $$\nabla\cdot f(v)=\sum_{e} f(e).$$ So $\nabla\cdot \nabla F(v)=\sum_{xy} c(x, y)(F(y)-F(x))$.