Total Curvature of Graphs after Milnor and Euler

TL;DR

This paper introduces a new concept called net total curvature for finite graphs in Rn, combining ideas from Milnor and Euler to analyze the complexity of spatial graphs through integral geometric methods.

Contribution

It defines net total curvature for graphs, linking local crookedness and Eulerian circuits, enabling analysis of graph complexity via geometric and topological properties.

Findings

Provides bounds for total curvature on isotopy classes

Demonstrates net total curvature as a measure of graph complexity

Offers a new functional for analyzing spatial graphs

Abstract

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in Rn, and investigate its properties. Two guiding principles are given by Milnor's way of measuring the local crookedness of a Jordan curve via a Crofton-type formula, and by considering the double cover of a given graph as an Eulerian circuit. The strength of combining these ideas in defining the curvature functional is (1) it allows us to interpret the singular/non-eulidean behavior at the vertices of the graph as a superposition of vertices of a 1-dimensional manifold, and thus (2) one can compute the total curvature for a wide range of graphs by contrasting local and global properties of the graph utilizing the integral geometric representation of the curvature. A collection of results on upper/lower bounds of the total curvature on isotopy/homeomorphism classes of embeddings is presented, which in turn demonstrates the effectiveness of net total curvature as a new functional measuring complexity of spatial graphs in differential-geometric terms.