Mobius energy of graphs

TL;DR

This paper introduces the Mobius energy for embedded graphs, explores its invariance under inversions, and analyzes critical configurations and symmetric constructions in three-dimensional space.

Contribution

It presents the first formulation of Mobius energy for graphs, investigates its properties, and discusses methods for constructing symmetric critical configurations.

Findings

Mobius energy is invariant under inversions in 3D space.

Critical configurations are characterized for vertices of degree less than five.

Techniques for constructing symmetric toric embedded graphs with critical energy values.

Abstract

In the present paper we introduce Mobius energy for the embedded graphs and formulate its main properties. This energy is invariant under the action of the group generated by all inversions in three-dimensional real space. We study critical configurations for the angles at vertices of degree less than five, and discuss the techniques of construction of symmetric toric embedded graphs with critical values of Mobius energy.