Walk entropy and walk-regularity
| Autor: | Daniel Král, Kyle Kloster, Blair D. Sullivan |
|---|---|
| Rok vydání: | 2018 |
| Předmět: |
Numerical Analysis
Algebra and Number Theory Conjecture 0211 other engineering and technologies 021107 urban & regional planning 010103 numerical & computational mathematics 02 engineering and technology 01 natural sciences Combinatorics Mathematics::Probability FOS: Mathematics Discrete Mathematics and Combinatorics Entropy (information theory) Mathematics - Combinatorics Geometry and Topology Combinatorics (math.CO) 0101 mathematics 05C50 QA Mathematics Counterexample |
| Zdroj: | Linear Algebra and its Applications |
| ISSN: | 0024-3795 |
| DOI: | 10.1016/j.laa.2018.02.009 |
| Popis: | A graph is said to be walk-regular if, for each $\ell \geq 1$, every vertex is contained in the same number of closed walks of length $\ell$. We construct a $24$-vertex graph $H_4$ that is not walk-regular yet has maximized walk entropy, $S^V(H_4,\beta) = \log 24$, for some $\beta>0$. This graph is a counterexample to a conjecture of Benzi [Linear Algebra Appl.~443 (2014), 395--399, Conjecture 3.1]. We also show that there exist infinitely many temperatures $\beta_0>0$ so that $S^V(G,\beta_0)=\log n_G$ if and only if a graph $G$ is walk-regular. Comment: 7 pages, 1 figure |
| Databáze: | OpenAIRE |
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