{"id":319,"date":"2020-04-21T09:54:34","date_gmt":"2020-04-21T09:54:34","guid":{"rendered":"http:\/\/blogs.rollins.edu\/graphembeddings\/?page_id=319"},"modified":"2020-04-21T10:19:49","modified_gmt":"2020-04-21T10:19:49","slug":"list-of-equations","status":"publish","type":"page","link":"https:\/\/blogs.rollins.edu\/graphembeddings\/embedding-complete-bipartite-graphs-with-cayley-maps\/list-of-equations\/","title":{"rendered":"List of Equations"},"content":{"rendered":"\n\n\n

Euler Characteristic Formula:<\/strong> \"\chi = V - E + F\", where \"V\" is the number of vertices, \"E\" is the number of edges, and \"F\" is the number of faces.<\/p>\n\n\n\n

Genus Formula:<\/strong> \"2-2g=\chi\", where \"g\" is the genus.<\/p>\n\n\n\n

Equation 1.<\/strong> For each \"x_i \in \rho\", \"\lambda(x_i)=\rho(x_i^{-1})\".<\/p>\n\n\n\n

Equation 2.<\/strong> For \"n_e\", where \"n_e\" is an even integer, \"L(n_e,n_e)=(\frac{n_e}{2})^2-n_e+1\".<\/p>\n\n\n\n

Equation 3.<\/strong> For \"n_o\", where \"n_o\" is an odd integer, \"L(n_o,n_o)=\frac{n_o^2+3}{4}-n_o+1\".<\/p>\n\n\n\n

Equation 4. <\/strong>Let \"m\" be the order of the sum of the elements in \"F_j\". The total number of edges, \"t\", of the face produced by \"\lambda_j\" is \"(t = m|F_j|)\".<\/p>\n\n\n\n

Equation 5.<\/strong> The number of occurrences, \"c\", of each face is \"c=\frac{2p|F_j|}{t} = \frac{2p}{m}\", where \"m\" is the order of the sum of the elements in \"F_j\".<\/p>\n\n\n\n

Equation 6. <\/strong>The number of darts, \"d\", in \"C_M(Z_{2p}, \rho)\" is \"d=ct= pV\".<\/p>\n","protected":false},"excerpt":{"rendered":"

Euler Characteristic Formula: $\\chi = V – E + F$, where $V$ is the number of vertices, $E$ is the number of edges, and $F$ is the number of faces. Genus Formula: $2-2g=\\chi$, where $g$ is the genus. Equation 1. For each $x_i \\in \\rho$, $\\lambda(x_i)=\\rho(x_i^{-1})$. Equation 2. For $n_e$, where $n_e$ is an even […]<\/p>\n","protected":false},"author":5,"featured_media":0,"parent":97,"menu_order":8,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-319","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/319","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=319"}],"version-history":[{"count":7,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/319\/revisions"}],"predecessor-version":[{"id":366,"href":"https:\/\/blogs.rollins.edu\/graphembeddings\/wp-json\/wp\/v2\/pages\/319\/revisions\/366"}],"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=319"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}