  We consider a generalization of the diffusion equation on graphs. This
generalized diffusion equation gives rise to both normal and superdiffusive
processes on infinite one-dimensional graphs. The generalization is based on
the $k$-path Laplacian operators $L_{k}$, which account for the hop of a
diffusive particle to non-nearest neighbours in a graph. We first prove that
the $k$-path Laplacian operators are self-adjoint. Then, we study the
transformed $k$-path Laplacian operators using Laplace, factorial and Mellin
transforms. We prove that the generalized diffusion equation using the Laplace-
and factorial-transformed operators always produce normal diffusive processes
independently of the parameters of the transforms. More importantly, the
generalized diffusion equation using the Mellin-transformed $k$-path Laplacians
$\sum_{k=1}^{\infty}k^{-s}L_{k}$ produces superdiffusive processes when
$1<s<3$.
