Shortest Paths in Distance-regular Graphs

We aim here at introducing a new point of view of the Laplacian of a graph, Γ. With this purpose in mind, we consider L as a kernel on the finite space V (Γ), in the context of the Potential Theory. Then we prove that L is a nice kernel, since it verifies some fundamental properties as maximum and energy principles and the equilibrium principle on any proper subset of V (Γ). If Γ is a proper set of a suitable host graph, then the equilibrium problem for Γ can be solved and the number of the different components of its equilibrium measure leads to a bound on the diameter of Γ. In particular, we obtain the structure of the shortest paths of a distance-regular graph. As a consequence, we find the intersection array in terms of the equilibrium measure. Finally, we give a new characterization of strongly regular graphs.