### Abstract

The notion of walk entropy SV(G,β)SV(G,β) for a graph G at the inverse temperature β was put forward recently by Estrada et al. (2014) [7]. It was further proved by Benzi [1] that a graph is walk-regular if and only if its walk entropy is maximum for all temperatures β∈Iβ∈I, where I is a set of real numbers containing at least an accumulation point. Benzi [1] conjectured that walk regularity can be characterized by the walk entropy if and only if there is a β>0β>0 such that SV(G,β)SV(G,β) is maximum. Here we prove that a graph is walk regular if and only if the View the MathML sourceSV(G,β=1)=lnn. We also prove that if the graph is regular but not walk-regular View the MathML sourceSV(G,β)<lnn for every β>0β>0 and View the MathML sourcelimβ→0SV(G,β)=lnn=limβ→∞SV(G,β). If the graph is not regular then View the MathML sourceSV(G,β)≤lnn−ϵ for every β>0β>0, for some ϵ>0ϵ>0.

Original language | English |
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Pages (from-to) | 542-547 |

Number of pages | 6 |

Journal | Linear Algebra and its Applications |

Volume | 458 |

Early online date | 9 Jul 2014 |

DOIs | |

Publication status | Published - 1 Oct 2014 |

### Keywords

- walk-regularity
- graph walks
- graph entropies

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## Cite this

Estrada, E., & de la Pena, J. A. (2014). Maximum walk entropy implies walk regularity.

*Linear Algebra and its Applications*,*458*, 542-547. https://doi.org/10.1016/j.laa.2014.06.030