{"id":63,"date":"2020-04-16T02:59:42","date_gmt":"2020-04-16T02:59:42","guid":{"rendered":"http:\/\/blogs.rollins.edu\/graphembeddings\/?page_id=63"},"modified":"2020-05-05T09:57:58","modified_gmt":"2020-05-05T09:57:58","slug":"cayley-graphs-cayley-maps","status":"publish","type":"page","link":"https:\/\/blogs.rollins.edu\/graphembeddings\/embedding-complete-bipartite-graphs-with-cayley-maps\/cayley-graphs-cayley-maps\/","title":{"rendered":"Cayley Graphs & Cayley Maps"},"content":{"rendered":"\n\n\n

Cayley Graphs<\/strong>
Cayley Graphs are a way of representing groups. Let \"G\" be a group with generating set \"S\". The Cayley Graph, \"C_G(G,S)\", is a graph where all members of \"G\" are vertices, and two vertices, \"v\" and \"w\", are adjacent if \"w=vs\" or \"v=ws\" for some \"s \in S\". <\/p>\n\n\n\n

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Cayley Graph \"C_G(Z_{10},\{1,3,5\})\", drawn two ways.<\/strong> The red lines represent the vertices that are connected by the 1 in the generating set, the blue lines represent the vertices connected by the 3, and the green lines represent the vertices connected by the 5. Notice that \"C_G(Z_{10},{1,3,5})\" is the same as \"K_{5,5}\".<\/figcaption><\/figure><\/div>\n\n\n\n

Cayley Maps<\/strong>
Let \"S'\" be a closed generating set of \"G\". The Cayley Map \"C_M(G,\rho)\" is an embedding of \"C_G(G,S')\" onto an orientable surface, where \"\rho=(x_0,\dots, x_{n-1})\" is a rotation (cyclic permutation) of \"S'\".<\/p>\n\n\n\n

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Cayley Map \"C_M(G,(x_0,\dots, x_{n-1}))\" where \"a \in G\". Notice that the elements in the rotation are assigned to the branches of the Cayley Map in counterclockwise order<\/figcaption><\/figure><\/div>\n\n\n\n

<\/p>\n","protected":false},"excerpt":{"rendered":"

Cayley GraphsCayley Graphs are a way of representing groups. Let $G$ be a group with generating set $S$. The Cayley Graph, $C_G(G,S)$, is a graph where all members of $G$ are vertices, and two vertices, $v$ and $w$, are adjacent if $w=vs$ or $v=ws$ for some $s \\in S$. Cayley MapsLet $S’$ be a closed […]<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":97,"menu_order":3,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-63","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/63","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/users\/5"}],"replies":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/comments?post=63"}],"version-history":[{"count":61,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/63\/revisions"}],"predecessor-version":[{"id":419,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/63\/revisions\/419"}],"up":[{"embeddable":true,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/97"}],"wp:attachment":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/media?parent=63"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}