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.

翻译：图表示是将几何图形从平面推广到更高维度的结果。Tutte引入的一种优化图形绘制性质的方法是使其能量最小化。我们探讨了球面图表示中的这种最小化问题，其中顶点位于单位球面上，且原点为其重心。我们提出了原始和对偶半定规划，可用于求解最小化能量的球面图表示。对于给定图$G$，我们将此规划的最优值记为$\rho(G)$。该值与$G$的邻接矩阵的第二大特征值（记为$\lambda_2$）相关。我们证明，对于正则图$G$，有$\rho(G) \leq \frac{\lambda_{2}}{2} \cdot v(G)$，且等式成立当且仅当$\lambda_2$的特征子空间包含一个球面1-设计。此外，若$G$为随机$d$-正则图，则渐近几乎必然有$\rho(G)=\left(\sqrt{(d-1)} +o(1)\right)\cdot v(G)$。