Potential Theory on graphs and embedding inequalities

Let $\Gamma$ be a finite graph with a vertex set $\mathcal{V}(\Gamma)$ and edge set $E(\Gamma)$. Given a function $K: \mathcal{V}(\Gamma) \times \mathcal{V}(\Gamma) \to \mathbb R_+$ and a measure $\mu: \mathcal{V}(\Gamma) \to \mathbb R_+$ we can define the potential
$$\mathbf V_K^{\mu}(\alpha)=\sum_{\beta\in\mathcal{V}(\Gamma)} K(\alpha,\beta)\mu(\beta).$$
       In particular, the  weighted Hardy potential on a rooted directed graph without directed cycles is given by
$$K_w(\alpha,\beta)=\sum_{e\ge \alpha,\beta} w(e),$$
      where $w: E(\Gamma) \to \mathbb R_+$ is a finite positive weight function defined on edges. We will present several examples of graph potentials, give a short overview of their connections to the problems in continuum and discuss some related properties, specifically capacitary inequalities and descriptions of trace (Carleson) measures for weighted Hardy embeddings on products of trees.