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

Total Curvature of Graphs after Milnor and Euler