Total curvature of graphs after Milnor and Euler

Abstract

We define a new notion of total curvature, called net total curvature, for finite graphs embedded in ℝ n, and investigate its properties. Two guiding principles are given by Milnor's way of measuring using a local Croftontype 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 that it allows us to interpret the singular/noneuclidean behavior at the vertices of the graph as a superposition of vertices of a 1-dimensional manifold, so that 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. © 2012 by Pacific Journal of Mathematics.