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 inR n , and investigate its properties. Two guiding principles ar e given by Milnor's way of measuring the local crookedness of a Jorda n 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 f unctional 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 curva- ture for a wide range of graphs by contrasting local and global properties of the graph utilizing the integral geometric representation of the cur vature. A collection of results on upper/lower bounds of the total curvature on isotopy/homeomorphism classes of embed- dings 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.