Entropy and the growth rate of universal covering trees
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909001819619328 |
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| author | Eisner, Idan Hoory, Shlomo |
| author_facet | Eisner, Idan Hoory, Shlomo |
| contents | This work studies the relation between two graph parameters, $ρ$ and $Λ$. For an undirected graph $G$, $ρ(G)$ is the growth rate of its universal covering tree, while $Λ(G)$ is a weighted geometric average of the vertex degree minus one, corresponding to the rate of entropy growth for the non-backtracking random walk (NBRW). It is well known that $ρ(G) \geq Λ(G)$ for all graphs, and that graphs with $ρ=Λ$ exhibit some special properties. In this work we derive an easy to check, necessary and sufficient condition for the equality to hold. Furthermore, we show that the variance of the number of random bits used by a length $\ell$ NBRW is $O(1)$ if $ρ= Λ$ and $Ω(\ell)$ if $ρ> Λ$. As a consequence we exhibit infinitely many non-trivial examples of graphs with $ρ= Λ$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_10337 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Entropy and the growth rate of universal covering trees Eisner, Idan Hoory, Shlomo Combinatorics 05C81 This work studies the relation between two graph parameters, $ρ$ and $Λ$. For an undirected graph $G$, $ρ(G)$ is the growth rate of its universal covering tree, while $Λ(G)$ is a weighted geometric average of the vertex degree minus one, corresponding to the rate of entropy growth for the non-backtracking random walk (NBRW). It is well known that $ρ(G) \geq Λ(G)$ for all graphs, and that graphs with $ρ=Λ$ exhibit some special properties. In this work we derive an easy to check, necessary and sufficient condition for the equality to hold. Furthermore, we show that the variance of the number of random bits used by a length $\ell$ NBRW is $O(1)$ if $ρ= Λ$ and $Ω(\ell)$ if $ρ> Λ$. As a consequence we exhibit infinitely many non-trivial examples of graphs with $ρ= Λ$. |
| title | Entropy and the growth rate of universal covering trees |
| topic | Combinatorics 05C81 |
| url | https://arxiv.org/abs/2410.10337 |