  Graph representations are the generalization of geometric graph drawings from
the plane to higher dimensions. A method introduced by Tutte to optimize
properties of graph drawings is to minimize their energy. We explore this
minimization for spherical graph representations, where the vertices lie on a
unit sphere such that the origin is their barycentre. We present a primal and
dual semidefinite program which can be used to find such a spherical graph
representation minimizing the energy. We denote the optimal value of this
program by $\rho(G)$ for a given graph $G$. The value turns out to be related
to the second largest eigenvalue of the adjacency matrix of $G$, which we
denote by $\lambda_2$. We show that for $G$ regular, $\rho(G) \leq
\frac{\lambda_{2}}{2} \cdot v(G)$, and that equality holds if and only if the
$\lambda_{2}$ eigenspace contains a spherical 1-design. Moreover, if $G$ is a
random $d$-regular graph, $\rho(G)=\left(\sqrt{(d-1)} +o(1)\right)\cdot v(G)$,
asymptotically almost surely.
