A Goodness-of-Fit Test for Uniformity on Distance-Regular Graphs with Applications to Grouped Directional Data

One of the main tasks in the field of directional statistics is to build goodness-of-fit tests for uniformity on circles, spheres or more abstract manifolds. We discuss a goodness-of-fit test for uniformity on a distance-regular graph. Our main tool is the pseudo-inverse of the combinatorial Laplacian, for which we give explicit expressions in terms of the intersection array of the graph. Such a test can be used as a test for uniformity of data from the circle or the sphere, grouped on the tiles of a regular tessellation associated with some finite group of isometries. We describe the cases of Platonic graphs and provide examples based on real circular and spherical data, vectorial or axial.