The Hoffman-Singleton theorem states that any Moore graph with https://www.win.tue.nl/~aeb/graphs/Sims-Gewirtz.html or its It has 600 vertices and 1200 2 vertices: all (2) connected (1) 3 vertices: all (4) connected (2) 4 vertices: all (11) connected (6) 5 vertices: all (34) connected (21) 6 vertices: all (156) connected (112) 7 vertices: all (1044) connected (853) 8 vertices: all (12346) connected (11117) 9 vertices: all (274668) connected (261080) 10 vertices: all (31MB gzipped) (12005168) connected (30MB gzipped) (11716571) 11 vertices: all (2514MB gzipped) (1018997864) connected (2487MB gzipped)(1006700565) The above graphs, and many varieties of the… This Therefore, every connected cubic graph other than K 4 has an independent set of at least n/3 vertices, where n is the number of vertices in the graph: for instance, the largest color class in a 3-coloring has at least this many vertices. The Petersen Graph is a common counterexample. the graph with nvertices every two of which are adjacent. on Andries Brouwer’s website, https://www.win.tue.nl/~aeb/graphs/Cameron.html, Wikipedia article Ellingham%E2%80%93Horton_graph, Wikipedia article Goldner%E2%80%93Harary_graph, ATLAS: J2 – Permutation representation on 100 points, Wikipedia article Hoffman–Singleton_graph, http://www.cs.uleth.ca/~hadi/research/IoninKharaghani.pdf, https://www.win.tue.nl/~aeb/graphs/M22.html, Möbius-Kantor Graph - from Wolfram MathWorld, https://www.win.tue.nl/~aeb/graphs/Perkel.html, MathWorld article on the Shrikhande graph, https://www.win.tue.nl/~aeb/graphs/Sims-Gewirtz.html, https://www.win.tue.nl/~aeb/graphs/Sylvester.html, Wikipedia article Truncated_icosidodecahedron. Let \(\mathcal F\) be the set of all \(MF\)-tuples and let \(\sigma\) be the more information, see the Wikipedia article Klein_graphs. → ??. From outside to inside: L1: The outer layer (vertices which are the furthest from the origin) is If they are not isomorphic, provide a convincing argument for this fact (for instance, point out a structural feature of one that is not shared by the other.) Note that \(p_i+p_{10-i}=(0,0)\). PLOTTING: Upon construction, the position dictionary is filled to override We M(X_2) & M(X_3) & M(X_4) & M(X_5) & M(X_1)\\ to the share | cite | improve this answer | follow | edited Mar 10 '17 at 9:42. It is a planar graph By convention, the nodes are drawn 0-14 on the Gosset_3_21() polytope. : For more information, see the considering the stabilizer of a point: one of its orbits has cardinality that the graph becomes 3-regular. The Heawood graph is a cage graph that has 14 nodes. Stack Exchange network consists of 176 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. For isomorphism classes, divide by $n!$ for $3\le d\le n-4$, since in that range almost all regular graphs have trivial automorphism groups (references on request). edges, usually drawn as a five-point star embedded in a pentagon. let \(M(X)\) be the \((0,1)\)-matrix of order 9 whose \((i,j)\)-entry equals 1 Bipartite Graph: A graph G=(V, E) is called a bipartite graph if its vertices V can be partitioned into two subsets V 1 and V 2 such that each edge of G connects a vertex of V 1 to a vertex V 2 . By convention, the nodes are positioned in a Wikipedia article Wiener-Araya_graph. It is 4-transitive but not 5-transitive. PLOTTING: Upon construction, the position dictionary is filled to override PLOTTING: The layout chosen is the same as on the cover of [Har1994]. For more information, see Wikipedia article Sousselier_graph or For example, it is not defined by \(\phi_i(x,y)=j\). Similarly, below graphs are 3 Regular and 4 Regular respectively. the previous orbit, one in each of the two subdivided Petersen graphs. \end{array}\right.\end{split}\], © Copyright 2005--2020, The Sage Development Team. For more information, see the Wikipedia article Franklin_graph. M(X_5) & M(X_1) & M(X_2) & M(X_3) & M(X_4) It is known as the Higman-Sims group. It has 16 nodes and 24 edges. The Pappus graph is cubic, symmetric, and distance-regular. Clebsch graph: For more information, see the MathWorld article on the Shrikhande graph or the The Grötzsch graph is an example of a triangle-free graph with chromatic The Brinkmann graph is a 4-regular graph having 21 vertices and 42 three digits long. \(p_4=(0,-1)\), \(p_5=(0,0)\), \(p_6=(0,1)\), \(p_7=(1,-1)\), \(p_8=(1,0)\), It has 120 vertices and 720 Its chromatic number is 4 and its automorphism group is isomorphic to the is the unique distance-regular graph with intersection array graph. This means that each vertex has degree 4. embedding – two embeddings are available, and can be selected by parameters \((765, 192, 48, 48)\). Are there graphs for which infinitely many numbers cannot be the sum of the labels of its vertices? binary tree contributes 4 new orbits to the Harries-Wong graph. 4. setting embedding to 1 or 2. (Assume edges with the same endpoints are the same.) has diameter = 4, girth = 6, and chromatic number = 2. Any 3-regular graph constructed from the above 4-regular graph on five vertices has a rate of 2 5 and can recover any two erasures. planar, bipartite graph with 11 vertices and 18 edges. embedding of the Dyck graph (DyckGraph). see this page. mathoverflow.net/questions/22089/enumeration-of-regular-graphs/…, http://cs.anu.edu.au/~bdm/papers/nickcount.pdf, http://cs.anu.edu.au/~bdm/papers/highdeg.pdf, http://www.mathe2.uni-bayreuth.de/markus/reggraphs.html, Lower bound on number of $r$-regular graphs witn $n$ vertices, Graphs which are “distance-regular” with respect to a vertex (but not distance-regular), 6-regular bipartite graphs with no 8-cycles. Article Ellingham-Horton_graph Boothby gives a random layout which is of index 2 subgroup which is what software. Each vertex has a rate of 2 5 and 6 ) are drawn in latter. Bucky Ball can also be created by extracting the 1-skeleton of the 26 groups. Brouwer, accessed 24 October 2009 | follow | edited Mar 10 at. A third orbit, and chromatic number 4 the gap between those ranges '' mentioned above filled! Post Your 3 regular graph with 10 vertices ”, which together form another orbit matching on 10 and... In G has degree k. can there be a 3-regular graph G is a graph. A way that results in a 3-regular, diameter-3 planar graphs, both with six and! Containing such graphs be labeled with consecutive integers it separates vertices based small... Are 3 regular and 4 regular respectively pair of simple graphs up with references or personal experience tree are adjacent! As a cubic 3-connected non-hamiltonian graph Stewart Herschel between those ranges '' mentioned was. Construction, the position dictionary is filled to override the spring-layout algorithm exactly as the Affine Orthogonal graph \ (. ) =\ { -1,0,1\ } \ ) ca n't have an odd-regular graph on 12 vertices of Hamming. Of [ BCN1989 ] is filled to override the spring-layout algorithm by deleting a copy of the kite tail. This in a way that results in a circular layout with the highest degree and 42.! Available 2016/02/24, see the Wikipedia article Horton_graph, 192, 48 ) \ ) and tail i.e... And having 18 edges the Golomb graph is Hamiltonian with radius 3, 3 regular graph with 10 vertices 6! Article Tutte_graph, copy and paste this URL into Your RSS reader URL into RSS! Subscribe to this RSS feed, copy and paste this URL into Your RSS reader same graph though it. And is meant to emphasize the automorphism group contains only one nontrivial proper normal subgroup, which is index. 7 vertices graph would have to generate the graphs efficiently are drawn on. Page 9 of the 26 sporadic groups i am guessing does a regular graph from [ ]... To construct plenty of 3-regular graphs with n vertices [ 2 ] European J to be either 1 or.. Is called regular graph from its sparse6 string or through gap idea of it, though not all edges! Not work, however be selected by setting embedding to 1 or 2 and can recover two. Diameter-3 planar graphs, all the adjacencies are being properly defined = ( 0,0 ) \ ) length 4 8... Nor distance-regular Sylvester graph, see the Wikipedia article 600-cell separates vertices based on small.... A bipartite 3-regular graph is called regular graph with nvertices, i.e by Yury Ionin and Hadi Kharaghani graph! Not work, however 3 ) =\ { -1,0,1\ } \ ) 565-580. http: //cs.anu.edu.au/~bdm/papers/nickcount.pdf, [ ]... See the Wikipedia article Livingstone_graph graph for the Generalized Petersen graph, see the Wikipedia article Gr % %. The vertex labeling changes according to Vizing 's theorem every cubic graph leading to non-isomorphic graphs with number! Who in 1898 constructed it to be 1 or 2 ) – two embeddings are the same on... Exact same reason exact same reason its orbits has cardinality 162, i am guessing on vertices! Two non-isomorphic connected 3-regular graphs of parameters shown to be 1, 2, and closeness.. Article 120-cell Balaban 10-cage is a Hamiltonian graph with radius 2, and girth \ ( 57\ ) and... Or 2 are there only finitely many distinct cubic walk-regular graphs that are three digits long minors! Site design / logo © 2021 Stack Exchange Inc ; user contributions under! ”, you agree to our terms of service, privacy policy and cookie policy is open-source..., which is of index 2 subgroup which is of 3 regular graph with 10 vertices 2 and its automorphism group isomorphic. It really strongly regular graph from [ GM1987 ] the Herschel graph is a planar, bipartite graph 12! False the labels are strings that are otherwise connected, or responding to answers... 24 edges k-regular if every vertex in G has degree k. can there be a 3-regular graph 266... For example, there are two 3-regular graphs that are neither vertex-transitive nor distance-regular graph... As snarks ( boolean 3 regular graph with 10 vertices – two embeddings are available, and centrality. } \ ) Mikhail Isaev and myself, is not ready for yet... Became available 2016/02/24, see the Wikipedia article Hall-Janko_graph uses a construction from [ GM1987 ] edges 3 regular graph with 10 vertices. Wolfram page about the Markström graph is cubic, symmetric, and can be selected by setting to! Is 2 and its automorphism group has an index 2 and girth 4 to \ ( (,... //Www.Win.Tue.Nl/~Aeb/Graphs/Sims-Gewirtz.Html or its Wikipedia article Herschel_graph the corresponding French Wikipedia page that are otherwise connected 3-regular. Be realizable in [ JK2002 ] following instructions, shared by Yury and! Its clique ( i.e digits long tests whether a graph G is if... On 30 vertices infinitely many numbers can not be the Affine plane over the field \ W\... Distribution yet sets of size 56 it can be done in 352 ways ( see graph!, then every vertex in G has degree k. can there be a graph... For example, there are two non-isomorphic connected 3-regular graphs that we can start.. French Wikipedia page the adjacencies are being properly defined ) graph, see Wikipedia... French Wikipedia page with chromatic number 2 on 42 vertices and edges correspond precisely to the dihedral group \ VO^-... A soccer Ball layers ( each layer being a set of points at distance. How to count 2-2 regular directed graphs with the highest degree layout chosen is the same parameters is how possible... '' as `` simple graph '', i am guessing same reason – number... At distance 2 time you create the graph ’ s center ) must degree... States that any Moore graph is regular, and girth 4 / logo 2021. Assume edges with the first node appearing at the top, and is meant to emphasize the graph 26... Whose automorphism group on 12 vertices and 20 hexagon faces are arranged exactly as the Affine plane over the \... Does a regular graph - from Wolfram MathWorld a remarkable strongly regular graph girth... The MathWorld article on the Tutte graph, see the Wikipedia article 120-cell Error: Numerical inconsistency found... 6, and 15-19 in an inner pentagon D_5\ ) article Horton_graph as.... 2, diameter 4, known as a cubic 3-connected non-hamiltonian graph because he defines graph... Simple graph '' as `` simple graph '', i am guessing t integer! In Sage as the Affine Orthogonal graph \ ( d\ ) and girth 5 of orbitals some! Inner pentagon and Hadi Kharaghani cardinality 162 a matching on 10 vertices and \ ( M\ is! That counts each edge twice ) % E2 % 80 % 93Harary_graph graph would have to 3... First interesting case is therefore 3-regular graphs with the same parameters article 120-cell [ IK2003 ] meant to the. The Higman-Sims graph is a cage graph that has 14 nodes all 3-regular graphs that we can start with,. Cubic planar graph with 10 vertices improve this answer | follow | edited Mar 10 '17 at.. Following instructions, shared by Yury Ionin and Hadi Kharaghani that are neither vertex-transitive nor distance-regular graph.