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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. )
added pointer to:
made explicit the “word metric” and added the redirect;
added pointer to:
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.
added pointer to:
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
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
and pointer to:
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