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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2411.18886 |
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| _version_ | 1866910719700631552 |
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| author | Mishura, Teddy |
| author_facet | Mishura, Teddy |
| contents | We present a characterization of Robinsonian $L^p$ graphons for $p > 5$. Each $L^p$ graphon $w$ is the limit object of a sequence of edge density-normalized simple graphs $\{G_n/\|G_n\|_1\}$ under the cut distance $δ_{\Box}$. A graphon $w$ is Robinson if it satisfies the Robinson property: if $x\leq y\leq z$, then $w(x,z)\leq \min\{w(x,y),w(y,z)\}$, and it is Robinsonian if $δ_{\Box}(w,u)=0$ for some Robinson $u$. In previous work, the author and collaborators introduced a graphon parameter $Λ$ that recognizes the Robinson property, where $Λ(w) = 0$ precisely when $w$ is Robinson. Using functional analytic arguments, we show here that for $p > 5$, the Robinsonian $L^p$ graphons $w$ are precisely those that are the cut distance limit object of graphs $G_n$ such that $Λ(G_n/\|G_n\|_1) \to 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_18886 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Partial Characterization of Robinsonian $L^p$ Graphons Mishura, Teddy Combinatorics 05C50 (Primary) 47B47, 54C08 (Secondary) We present a characterization of Robinsonian $L^p$ graphons for $p > 5$. Each $L^p$ graphon $w$ is the limit object of a sequence of edge density-normalized simple graphs $\{G_n/\|G_n\|_1\}$ under the cut distance $δ_{\Box}$. A graphon $w$ is Robinson if it satisfies the Robinson property: if $x\leq y\leq z$, then $w(x,z)\leq \min\{w(x,y),w(y,z)\}$, and it is Robinsonian if $δ_{\Box}(w,u)=0$ for some Robinson $u$. In previous work, the author and collaborators introduced a graphon parameter $Λ$ that recognizes the Robinson property, where $Λ(w) = 0$ precisely when $w$ is Robinson. Using functional analytic arguments, we show here that for $p > 5$, the Robinsonian $L^p$ graphons $w$ are precisely those that are the cut distance limit object of graphs $G_n$ such that $Λ(G_n/\|G_n\|_1) \to 0$. |
| title | A Partial Characterization of Robinsonian $L^p$ Graphons |
| topic | Combinatorics 05C50 (Primary) 47B47, 54C08 (Secondary) |
| url | https://arxiv.org/abs/2411.18886 |