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1. • CommentRowNumber1.
   • CommentAuthorTim_Porter
   • CommentTimeJul 14th 2018
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexPage created, but author did not leave any comments. <a href="https://ncatlab.org/nlab/revision/Cayley+graph/1">v1</a>, <a href="https://ncatlab.org/nlab/show/Cayley+graph">current</a>

   Page created, but author did not leave any comments.

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeJul 15th 2018
   • (edited Jul 15th 2018)
   • PermaLink
   Author: Tim_Porter
   Format: MarkdownItexI have been tweeking the wording to try and get across the link between the combinatorial group theory of the presentation and the algebraic topological encoding in the Cayley graph. It would be good to have a picture of the $S_3$ example, but I could not get the code to work so would ask someone who understands drawing such diagrams here better than I do to help. (I have various forms of the code, xypic, etc. )

   I have been tweeking the wording to try and get across the link between the combinatorial group theory of the presentation and the algebraic topological encoding in the Cayley graph. It would be good to have a picture of the $S_3$ example, but I could not get the code to work so would ask someone who understands drawing such diagrams here better than I do to help. (I have various forms of the code, xypic, etc. )

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 17th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Cornelia Druţu]], [[Michael Kapovich]], Section 7.9 of: *Geometric group theory*, Colloquium Publications **63**, AMS 2018 ([ISBN:978-1-4704-1104-6](https://bookstore.ams.org/coll-63/), [pdf](https://courses.maths.ox.ac.uk/node/view_material/35649)) <a href="https://ncatlab.org/nlab/revision/diff/Cayley+graph/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+graph/6">v6</a>, <a href="https://ncatlab.org/nlab/show/Cayley+graph">current</a>

   added pointer to:

   • Cornelia Druţu, Michael Kapovich, Section 7.9 of: Geometric group theory, Colloquium Publications 63, AMS 2018 (ISBN:978-1-4704-1104-6, pdf)

   diff, v6, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeApr 17th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexmade explicit the "word metric" and added the redirect; added pointer to: * [[Cornelia Druţu]], [[Michael Kapovich]], Section 7.9 of: *Geometric group theory*, Colloquium Publications **63**, AMS 2018 ([ISBN:978-1-4704-1104-6](https://bookstore.ams.org/coll-63/), [pdf](https://courses.maths.ox.ac.uk/node/view_material/35649)) <a href="https://ncatlab.org/nlab/revision/diff/Cayley+graph/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+graph/6">v6</a>, <a href="https://ncatlab.org/nlab/show/Cayley+graph">current</a>

   made explicit the “word metric” and added the redirect;

   added pointer to:

   • Cornelia Druţu, Michael Kapovich, Section 7.9 of: Geometric group theory, Colloquium Publications 63, AMS 2018 (ISBN:978-1-4704-1104-6, pdf)

   diff, v6, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeApr 17th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexmade explicit the example of the symmetric group: *** The [[symmetric group]] $Sym(n)$ may be [[finitely generated group|generated]] from * all [[transposition permutations]] -- the corresponding Cayley [[graph distance]] is the original *[[Cayley distance]]*; * the *adjacent* [[transpositions]] -- the corresponding Cayley [[graph distance]] is known as the *[[Kendall tau distance]]*. *** <a href="https://ncatlab.org/nlab/revision/diff/Cayley+graph/9">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+graph/9">v9</a>, <a href="https://ncatlab.org/nlab/show/Cayley+graph">current</a>

   made explicit the example of the symmetric group:

   ---

   The symmetric group $Sym(n)$ may be generated from

   • all transposition permutations – the corresponding Cayley graph distance is the original Cayley distance;

   • the adjacent transpositions – the corresponding Cayley graph distance is known as the Kendall tau distance.

   ---

   diff, v9, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 19th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Elena Konstantinova]], *Some problems on Cayley graphs*, Linear Algebra and its Applications Volume 429, Issues 11–12, 1 December 2008, Pages 2754-2769 ([doi:10.1016/j.laa.2008.05.010](https://doi.org/10.1016/j.laa.2008.05.010)) <a href="https://ncatlab.org/nlab/revision/diff/Cayley+graph/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+graph/10">v10</a>, <a href="https://ncatlab.org/nlab/show/Cayley+graph">current</a>

   added pointer to:

   • Elena Konstantinova, Some problems on Cayley graphs, Linear Algebra and its Applications Volume 429, Issues 11–12, 1 December 2008, Pages 2754-2769 (doi:10.1016/j.laa.2008.05.010)

   diff, v10, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeApr 20th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have tikz-ed the Cayley graph for $Sym(3)$, as an example, [here](https://ncatlab.org/nlab/show/Cayley+graph#CayleyGraphForSym3). I'll also give this its own little page now, *[[Cayley graph for Sym(3)]]*, in order to discuss a few more properties <a href="https://ncatlab.org/nlab/revision/diff/Cayley+graph/12">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+graph/12">v12</a>, <a href="https://ncatlab.org/nlab/show/Cayley+graph">current</a>

   I have tikz-ed the Cayley graph for $Sym(3)$, as an example, here.

   I’ll also give this its own little page now, Cayley graph for Sym(3), in order to discuss a few more properties

   diff, v12, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeApr 27th 2021
   • (edited Apr 27th 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added the following references on spectra of Cayley graphs ([here](https://ncatlab.org/nlab/show/Cayley+graph#ReferencesSpectra)): * [[László Lovász]], *Spectra of graphs with transitive groups*, Period. Math. Hungar., 6(2):191–195,1975 ([doi:10.1007/BF02018821](https://doi.org/10.1007/BF02018821), [pdf](https://web.cs.elte.hu/~lovasz/scans/spectra-of-graphs.pdf)) * [[László Babai]], *Spectra of Cayley graphs*, Journal of Combinatorial Theory, Series B Volume 27, Issue 2, October 1979, Pages 180-189 (<a href="https://doi.org/10.1016/0095-8956(79)90079-0">doi:10.1016/0095-8956(79)90079-0</a>) * Petteri Kaski, *Eigenvectors and spectra of Cayley graphs*, 2002 ([pdf](http://www.tcs.hut.fi/Studies/T-79.300/2002S/esitelmat/kaski_paper_020506.pdf)) * Xiaogang Liu, Sanming Zhou, *Eigenvalues of Cayley graphs* ([arXiv:1809.09829](https://arxiv.org/abs/1809.09829)) * Farzaneh Nowroozi, Modjtaba Ghorbani, *On the spectrum of Cayley graphs via character table*, Journal of Mathematical NanoScience, Volume 4, 1-2 (2014) ([doi:10.22061/jmns.2014.477](https://dx.doi.org/10.22061/jmns.2014.477) [pdf](https://journals.sru.ac.ir/article_477_166922038b274bc7128566c1e2ce4b98.pdf)) also, I have added ([here](https://ncatlab.org/nlab/show/Cayley+graph#CayleyGraphOfSym4)) a picture of the Cayley graph of $Sym(4)$ to the entry, taken from Kaski's article <a href="https://ncatlab.org/nlab/revision/diff/Cayley+graph/13">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+graph/13">v13</a>, <a href="https://ncatlab.org/nlab/show/Cayley+graph">current</a>

   I have added the following references on spectra of Cayley graphs (here):

   • László Lovász, Spectra of graphs with transitive groups, Period. Math. Hungar., 6(2):191–195,1975 (doi:10.1007/BF02018821, pdf)

   • László Babai, Spectra of Cayley graphs, Journal of Combinatorial Theory, Series B Volume 27, Issue 2, October 1979, Pages 180-189 (doi:10.1016/0095-8956(79)90079-0)

   • Petteri Kaski, Eigenvectors and spectra of Cayley graphs, 2002 (pdf)

   • Xiaogang Liu, Sanming Zhou, Eigenvalues of Cayley graphs (arXiv:1809.09829)

   • Farzaneh Nowroozi, Modjtaba Ghorbani, On the spectrum of Cayley graphs via character table, Journal of Mathematical NanoScience, Volume 4, 1-2 (2014) (doi:10.22061/jmns.2014.477 pdf)

   also, I have added (here) a picture of the Cayley graph of $Sym(4)$ to the entry, taken from Kaski’s article

   diff, v13, current

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeApr 27th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexand pointer to: * B. Foster-Greenwood, C. Kriloff, *Spectra of Cayley graphs of complex reflection groups*, J. Algebraic Combin., 44(1):33–57, 2016 ([arXiv:1502.07392](https://arxiv.org/abs/1502.07392), [doi:10.1007/s10801-015-0652-8](https://doi.org/10.1007/s10801-015-0652-8)) <a href="https://ncatlab.org/nlab/revision/diff/Cayley+graph/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+graph/14">v14</a>, <a href="https://ncatlab.org/nlab/show/Cayley+graph">current</a>

   and pointer to:

   • B. Foster-Greenwood, C. Kriloff, Spectra of Cayley graphs of complex reflection groups, J. Algebraic Combin., 44(1):33–57, 2016 (arXiv:1502.07392, doi:10.1007/s10801-015-0652-8)

   diff, v14, current

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2023
    • PermaLink
    Author: Urs
    Format: MarkdownItexadded pointer to: * Def. 6.3.4 in: *[[Symmetry]]* (2021) &lbrack;[pdf](https://unimath.github.io/SymmetryBook/book.pdf)&rbrack; <a href="https://ncatlab.org/nlab/revision/diff/Cayley+graph/18">diff</a>, <a href="https://ncatlab.org/nlab/revision/Cayley+graph/18">v18</a>, <a href="https://ncatlab.org/nlab/show/Cayley+graph">current</a>

    added pointer to:

    • Def. 6.3.4 in: Symmetry (2021) [pdf]

    diff, v18, current