1.6 and 1.24]).

Recapitulating the above:

1.  we can pick $\alpha = \beta = 1$ since this only scales a solution to ,

2.  we should pick $\gamma = 2$ to be able to calculate the rubber band contribution to efficiently,

3.  we should pick $\delta = d - 2$ to be able to approximate the repelling charge contribution to by treating clusters of charges as a single, heavier charge.

Inserting these constants we obtain the following expressions for $f$ and its partial derivatives: $$\begin{aligned}
f(x_1, \ldots, x_k)
& = \frac{1}{2} \sum_{j = 1}^l c_j \sum_{i \in e_j} \|x_i - z_j\|^2
+ \frac{1}{2} \sum_{j = 1}^k \sum_{\substack{i = 1 \\ i \neq j}}^k \frac{w_i \, w_j}{\|x_i - x_j\|^{d - 2}}, \label{eq:graphenergysimple} \\
\frac{\partial f(x_1, \ldots, x_k)}{\partial x_{m \, n}}
& = \sum_{\{j {\ |\ }n \in e_j\}} c_j \, (x_{m \, n} - z_{m \, j})
+ (d - 2) \sum_{\substack{i = 1 \\ i \neq n}}^k \frac{w_i \, w_n}{\|x_i - x_n\|^d} \, (x_{m \, i} - x_{m \, n}). \label{eq:graphenergygradientsimple}\end{aligned}$$

Connectedness
-------------

Before we start describing an algorithm to solve , we should take care to ensure that such a solution actually exists.

\[thm:disconnectedminimum\] Let $G = (V, E)$, 