2014/06/19 by Ernesto Estrada, Estrada, Ernesto, José Antonio de la Peña +1
Physics and Astronomy · #Advanced Thermodynamics and Statistical Mechanics #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum many-body systems #Theoretical and Computational Physics
paper · doi:10.48550/arxiv.1406.5056
openalex publication_date 2014/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The notion of walk entropy SV(G,β) for a graph G at the inverse temperature β was put forward recently by Estrada et al. (2014) \cite6. It was further proved by Benzi \cite1 that a graph is walk-regular if and only if its walk entropy is maximum for all temperatures β∈ I, where I is a set of real numbers containing at least an accumulation point. Benzi \cite1 conjectured that walk regularity can be characterized by the walk entropy if and only if there is a β>0, such that SV(G,β) is maximum. Here we prove that a graph is walk regular if and only if the SV(G,β=1)=ln n. We also prove that if the graph is regular but not walk-regular SV(G,β)0 and limβ→ 0 SV(G,β)=ln n=limβ→ ∞ SV(G,β). If the graph is not regular then SV(G,β) ≤ ln n-ε for every β>0, for some ε>0.