Abstract
The concept of tangent spaces that is utilized in physics to analyze geometric structure and motion in a smooth manifold are applied to finite simple connected graphs. This note develops the incidence structure of a graph that represents the tangent structure of a graph, develops vertex incidence spaces that are vector spaces at each vertex consisting of vectors tangent to that vertex, and reveals the connection operator of a graph. From the graph tangent structure perspective, an example of communication and information flow in a collaboration network that utilizes parallel transport is discussed along with a discussion of additional graph dimensions. Several concepts are left to this note's readers for further development such as possible tangent structure polynomials based on vertex degree and a possible application to knot theory. As with any new perspective, the full realm of applications is in the future.