  In this paper, we analyze the eigenfunctions of the edge-based Laplacian on a
graph and the relationship of these functions to random walks on the graph. We
commence by discussing the set of eigenfunctions supported at the vertices, and
demonstrate the relationship of these eigenfunctions to the classical random
walk on the graph. Then, from an analysis of functions supported only on the
interior of edges, we develop a method for explicitly calculating the
edge-interior eigenfunctions of the edge-based Laplacian. This reveals a
connection between the edge-based Laplacian and the adjacency matrix of
backtrackless random walk on the graph. The edge-based eigenfunctions therefore
correspond to some eigenfunctions of the normalised Hashimoto matrix.
