Elastic Graphs

TL;DR

This paper introduces elastic graphs with edge elasticity, analyzing when homotopy classes of maps minimize elastic energy, contributing to the understanding of energy-based graph mappings.

Contribution

It characterizes conditions under which homotopy classes of elastic graph maps minimize elastic energy, expanding the framework of energy-based graph mappings.

Findings

Identifies when a homotopy class decreases elastic energy

Provides a framework for energy minimization in graph mappings

Connects elastic energy concepts to graph homotopy classes

Abstract

An elastic graph is a graph with an elasticity associated to each edge. It may be viewed as a network made out of ideal rubber bands. If the rubber bands are stretched on a target space there is an elastic energy. We characterize when a homotopy class of maps from one elastic graph to another is loosening, i.e., decreases this elastic energy for all possible targets. This fits into a more general framework of energies for maps between graphs.