### Abstract

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*, which account for the hop of a diffusive particle to non-nearest neighbours in a graph. We first prove that the_{k}*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 Σ^{∞}_{k=1}*k*^{-s}*L*produces superdiffusive processes when 1 <_{k}*s*< 3.Original language | English |
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Pages (from-to) | 307-334 |

Number of pages | 28 |

Journal | Linear Algebra and its Applications |

Volume | 523 |

Early online date | 24 Feb 2017 |

DOIs | |

Publication status | Published - 15 Jun 2017 |

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### Keywords

- k-path Laplacian
- anomalous diffusion

### Cite this

Estrada, E., Hameed, E., Hatano, N., & Langer, M. (2017). Path Laplacian operators and superdiffusive processes on graphs. I. one-dimensional case.

*Linear Algebra and its Applications*,*523*, 307-334. https://doi.org/10.1016/j.laa.2017.02.027