In this talk we consider certain graphs that arise out of matching shapes or images. We begin by exploring how to define isometrically-invariant descriptors of shape neighborhoods and how these can be used to compute maps between shapes. We then analyze map networks with the goal of improving individual maps connecting the shapes. The fact that maps compose algebraically allows certain novel approaches that are not possible when edges encode only some proximity or similarity of the nodes. Finally we look at much larger networks of images connected via matching feature sets and look at how methods of spectral graph theory and algebraic topology can help both the construction and the simplification of such networks.