{"properties":[{"name":"Connected","aliases":[],"uid":"P000001","description":"A graph is connected if there is a path between every pair of vertices.\n\nSee {{wikipedia:Connectivity_(graph_theory)}}.","refs":[{"name":"Connectivity on Wikipedia","wikipedia":"Connectivity_(graph_theory)"}]},{"name":"Regular","aliases":[],"uid":"P000002","description":"A graph is regular if every vertex has the same degree.\nA $k$-regular graph is one in which every vertex has degree $k$;\na $3$-regular graph is also called cubic.\n\nSee {{wikipedia:Regular_graph}}.","refs":[{"name":"Regular graph on Wikipedia","wikipedia":"Regular_graph"}]},{"name":"Vertex-transitive","aliases":[],"uid":"P000003","description":"A graph is vertex-transitive if its automorphism group acts transitively\non its vertices: for any two vertices $u$ and $v$ there is an automorphism\nof the graph mapping $u$ to $v$.\n\nSee {{wikipedia:Vertex-transitive_graph}}.","refs":[{"name":"Vertex-transitive graph on Wikipedia","wikipedia":"Vertex-transitive_graph"}]}],"spaces":[{"name":"Petersen graph","aliases":["Kneser graph $K(5,2)$","Generalized Petersen graph $GP(5,2)$"],"uid":"S000001","description":"The Petersen graph is the graph on $10$ vertices and $15$ edges realized as\nthe Kneser graph $K(5,2)$: its vertices are the $2$-element subsets of a\n$5$-element set, with two vertices adjacent if and only if the corresponding\nsubsets are disjoint.\n\nEquivalently, it is the generalized Petersen graph $GP(5,2)$: take an outer\n$5$-cycle on vertices $u_0,\\dots,u_4$, an inner set of vertices\n$v_0,\\dots,v_4$ joined as the pentagram $v_i \\sim v_{i+2}$ (indices mod $5$),\nand spokes $u_i \\sim v_i$ connecting them.\n\nSee {{wikipedia:Petersen_graph}}.","refs":[{"name":"Petersen graph on Wikipedia","wikipedia":"Petersen_graph"}]}],"theorems":[{"when":{"kind":"atom","property":"P000003","value":true},"then":{"kind":"atom","property":"P000002","value":true},"uid":"T000001","description":"If the automorphism group acts transitively on the vertices, then all vertices\nlie in a single orbit and therefore share the same degree. Hence every\nvertex-transitive graph is regular.","refs":[]}],"traits":[{"space":"S000001","property":"P000001","value":true,"uid":"","description":"\n\nThe Petersen graph is connected; in fact it is $3$-connected, as noted in\n{{wikipedia:Petersen_graph}}.\n","refs":[{"name":"Petersen graph on Wikipedia","wikipedia":"Petersen_graph"}]},{"space":"S000001","property":"P000002","value":true,"uid":"","description":"\n\nThe Petersen graph is $3$-regular (cubic): every vertex of the Petersen graph has degree $3$.\n","refs":[]},{"space":"S000001","property":"P000003","value":true,"uid":"","description":"\n\nThe automorphism group of the Petersen graph is the symmetric group $S_5$,\nacting on the vertices as the $2$-element subsets of a $5$-element set.\nThis action is transitive, so the graph is vertex-transitive.\n\nThe isomorphism $\\mathrm{Aut} \\cong S_5$ is established in the proof without\nwords {{doi:10.4169/math.mag.89.4.267}}.\n","refs":[{"name":"Wood, The Automorphism Group of the Petersen Graph is Isomorphic to $S_5$ (proof without words)","doi":"10.4169/math.mag.89.4.267"}]}],"version":{"ref":"main","sha":"67009199d1192fb353dd4393ddbbac128ddb0165"}}