Maximum walk entropy implies walk regularity
Estrada, Ernesto and de la Pena, Jose Antonio (2014) Maximum walk entropy implies walk regularity. Linear Algebra and its Applications, 458. pp. 542547. ISSN 00243795 (https://doi.org/10.1016/j.laa.2014.06.030)
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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 walkregular 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 walkregular 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.


Item type: Article ID code: 50879 Dates: DateEvent1 October 2014Published9 July 2014Published Online18 June 2014AcceptedSubjects: Science > Mathematics > Probabilities. Mathematical statistics Department: Faculty of Science > Mathematics and Statistics Depositing user: Pure Administrator Date deposited: 23 Dec 2014 13:49 Last modified: 07 May 2024 01:01 URI: https://strathprints.strath.ac.uk/id/eprint/50879