On Minimizing the Energy of a Spherical Graph Representation

TL;DR

This paper investigates the energy minimization of spherical graph representations, linking the optimal energy to eigenvalues of the adjacency matrix, and provides bounds and asymptotic results for regular and random graphs.

Contribution

It introduces a semidefinite programming approach to minimize energy in spherical graph representations and establishes bounds related to eigenvalues, with asymptotic analysis for regular graphs.

Findings

The optimal energy value relates to the second largest eigenvalue of the adjacency matrix.

For regular graphs, an upper bound on energy is established based on eigenvalues.

Asymptotic behavior of energy for random regular graphs is characterized.

Abstract

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.