Combinatorics of d-path Laplacians on paths and distance-regular graphs

Abstract. We investigate the combinatorial and algebraic structure of d-path Laplacians on path graphs and distance-regular graphs. In particular, we derive explicit expressions for powers of the standard Laplacian in terms of d-adjacency matrices and d-path Laplacians on the infinite path graph, obtaining exact formulas involving binomial coefficients. For finite path graphs, we show that the same bulk structure persists away from the boundaries, while additional correction terms emerge due to endpoint effects. We further characterize classes of graphs for which powers of the Laplacian admit expansions in terms of distance matrices, proving that this property is naturally associated with distance-regular graphs and their Bose–Mesner algebra structure. In addition, we show that Mellin-, Laplace-, and factorial-transformed d-path Laplacians on the infinite path graph can be represented as functions of the standard Laplacian through continuous functional calculus. These results clarify the interplay between graph combinatorics, spectral theory, and nonlocal diffusion operators, contributing to the mathematical foundations of anomalous transport and superdiffusive processes on discrete structures.