Lower bounds for the isoperimetric numbers of random regular graphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2013
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| _version_ | 1866911266747973632 |
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| author | Kolesnik, Brett Wormald, Nick |
| author_facet | Kolesnik, Brett Wormald, Nick |
| contents | The vertex isoperimetric number of a graph $G=(V,E)$ is the minimum of the ratio $|\partial_{V}U|/|U|$ where $U$ ranges over all nonempty subsets of $V$ with $|U|/|V|\le u$ and $\partial_{V}U$ is the set of all vertices adjacent to $U$ but not in $U$. The analogously defined edge isoperimetric number---with $\partial_{V}U$ replaced by $\partial_{E}U$, the set of all edges with exactly one endpoint in $U$---has been studied extensively. Here we study random regular graphs. For the case $u=1/2$, we give asymptotically almost sure lower bounds for the vertex isoperimetric number for all $d\ge3$. Moreover, we obtain a lower bound on the asymptotics as $d\to\infty$. We also provide asymptotically almost sure lower bounds on $|\partial_{E}U|/|U|$ in terms of an upper bound on the size of $U$ and analyse the bounds as $d\to\infty$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1311_6555 |
| institution | arXiv |
| publishDate | 2013 |
| record_format | arxiv |
| spellingShingle | Lower bounds for the isoperimetric numbers of random regular graphs Kolesnik, Brett Wormald, Nick Combinatorics Probability The vertex isoperimetric number of a graph $G=(V,E)$ is the minimum of the ratio $|\partial_{V}U|/|U|$ where $U$ ranges over all nonempty subsets of $V$ with $|U|/|V|\le u$ and $\partial_{V}U$ is the set of all vertices adjacent to $U$ but not in $U$. The analogously defined edge isoperimetric number---with $\partial_{V}U$ replaced by $\partial_{E}U$, the set of all edges with exactly one endpoint in $U$---has been studied extensively. Here we study random regular graphs. For the case $u=1/2$, we give asymptotically almost sure lower bounds for the vertex isoperimetric number for all $d\ge3$. Moreover, we obtain a lower bound on the asymptotics as $d\to\infty$. We also provide asymptotically almost sure lower bounds on $|\partial_{E}U|/|U|$ in terms of an upper bound on the size of $U$ and analyse the bounds as $d\to\infty$. |
| title | Lower bounds for the isoperimetric numbers of random regular graphs |
| topic | Combinatorics Probability |
| url | https://arxiv.org/abs/1311.6555 |