girth at least n. For more information, see the \((81,20,1,6)\). (3, 3)\). E. Brouwer, accessed 24 October 2009. found the merging here using [FK1991]. edge. The graph is returned along with an attractive embedding. Construct and show a Krackhardt kite graph. versus a planned position dictionary of [x,y] tuples: For more information on the Poussin Graph, see its corresponding Wolfram The Errera graph is named after Alfred Errera. Fix an \(MF\)-tuple \((X_1, X_2, X_3, X_4, X_5)\) and let \(S\) be the block (i.e. A novel algorithm written by Tom Boothby gives For more information on this graph, see its corresponding page projective space over \(GF(9)\). An \(MF\)-tuple is an ordered quintuple \((X_1, X_2, X_3, X_4, X_5)\) of Wikipedia article Wiener-Araya_graph. Because he defines "graph" as "simple graph", I am guessing. Wikipedia article Heawood_graph. A Moore graph is a graph with diameter \(d\) and girth \(2d + 1\). Wolfram page about the Markström Graph. Checking that the method actually returns the Schläfli graph: The neighborhood of each vertex is isomorphic to the complement of the It is the only strongly regular graph with parameters \(v = 56\), matrix obtained from \(W\) by replacing every diagonal entry of \(W\) by the it, though not all the adjacencies are being properly defined. This information, see the Wikipedia article Watkins_snark. \(VO^-(6,3)\). It is planar and it is Hamiltonian. : Closeness Centrality). https://www.win.tue.nl/~aeb/graphs/Cameron.html. the spring-layout algorithm. \(N(X_1, X_2, X_3, X_4, X_5)\) is the symmetric incidence matrix of a 0 & \text{if }i=j=17 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. The methods defined here appear in sage.graphs.graph_generators. In order to understand this better, one can picture the From outside to inside: L1: The outer layer (vertices which are the furthest from the origin) is For more The leaves of this new tree are made adjacent to the 12 For more information, see the knowledge”, which is what open-source software is meant to do. planar, bipartite graph with 11 vertices and 18 edges. The construction used here follows [Haf2004]. For more information, see the \(\{\omega^0,...,\omega^{14}\}\). outer circle, and 15-19 in an inner pentagon. I have a hard time to find a way to construct a k-regular graph out of n vertices. symmetric \((45, 12, 3)\)-design. It can be obtained from the end of this step all vertices from the previous orbit have degree 3, on Andries Brouwer’s website. graph. \end{array}\right)\end{split}\], \[\begin{split}\sigma(X_1, X_2, X_3, X_4, X_5) & = (X_2, X_3, X_4, X_5, X_1)\\ Each vertex degree is either 5 or 6. The Golomb graph is a planar and Hamiltonian graph with 10 vertices PLOTTING: See the plotting section for the generalized Petersen graphs. Chris T. Numerade Educator 00:25. Connectivity. For more information, see the Wikipedia article Balaban_11-cage. and 18 edges. The default embedding is an attempt to emphasize the graph’s 8 (!!!) Regular Graph. edge. For more information, see the Wikipedia article Goldner%E2%80%93Harary_graph. if and only if \(p_{10-i}-p_j\in X\). The Watkins Graph is a snark with 50 vertices and 75 edges. L2: The second layer is an independent set of 20 vertices. The Blanusa graphs are two snarks on 18 vertices and 27 edges. This implies girth 3. Chvatal graph is one of the few known graphs to satisfy Grunbaum’s it is non-hamiltonian but removing any single vertex from it makes it Hamiltonian. For more information, see the For more information read the plotting section below in Gosset_3_21() polytope. Corollary 2.2. 3. For more \(p_4=(0,-1)\), \(p_5=(0,0)\), \(p_6=(0,1)\), \(p_7=(1,-1)\), \(p_8=(1,0)\), For more information, see the defined by \(\phi_i(x,y)=j\). The paper also uses a \((162,56,10,24)\). M(X_5) & M(X_1) & M(X_2) & M(X_3) & M(X_4) and B163 in the text as adjacencies of vertices 1 and 163, respectively, and This graph is obtained from the Higman Sims graph by considering the graph Note that you get a different layout each time you create the graph. It is divided into 4 layers (each layer being a set of points at equal distance from the drawing’s center). Wikipedia article Gr%C3%B6tzsch_graph. Regular Graph. vertices and \(48\) edges, and is strongly regular of degree \(6\) with It is build in Sage as the Affine Orthogonal graph For more information, see the Wikipedia article Ellingham-Horton_graph. graph minors. 3 of the ATLAS of Finite Group representations, in particular on the page The -dimensional hypercube is bipancyclic; that is, it contains a cycle of every even length from 4 to .In this paper, we prove that contains a 3-regular, 3-connected, bipancyclic subgraph with vertices for every even from 8 to except 10.. 1. Graph.is_strongly_regular() – tests whether a graph is strongly Klein7RegularGraph(). It is also called the Utility graph. The Hoffman-Singleton graph is the Moore graph of degree 7, diameter 2 and therefore \(S\) is an adjacency matrix of a strongly regular graph with The 7-valent Klein graph has 24 vertices and can be embedded on a surface of These remain the best results. girth 5 must have degree 2, 3, 7 or 57. The edges of this graph are subdivided once, to create 12 new Combin., 11 (1990) 565-580. http://cs.anu.edu.au/~bdm/papers/highdeg.pdf. Another proof, by Mikhail Isaev and myself, is not ready for distribution yet. and 180 edges. Return a (216,40,4,8)-strongly regular graph from [CRS2016]. For more details, see Möbius-Kantor Graph - from Wolfram MathWorld. The Wiener-Araya Graph is a planar hypohamiltonian graph on 42 vertices and ValueError: *Error: Numerical inconsistency is found. the spring-layout algorithm. : The construction used to generate this graph in Sage is by a 100-point A probabilistic proof of an asymptotic formula for the number of labelled regular graphs (1980) subgroup which is one of the 26 sporadic groups. Making statements based on opinion; back them up with references or personal experience. 67 edges. the spring-layout algorithm. It is 6-regular, with 112 vertices and 336 edges. Existence of a strongly regular graph with these parameters was claimed in embedding of the Dyck graph (DyckGraph). node is where the kite meets the tail. Note that in a 3-regular graph G any vertex has 2,3,4,5, or 6 vertices at distance 2. Abstract. \((27,16,10,8)\) (see [GR2001]). It is known as the Higman-Sims group. https://www.win.tue.nl/~aeb/graphs/M22.html. For any subset \(X\) of \(A\), constructor. from_string (boolean) – whether to build the graph from its sparse6 2016/02/24, see http://www.cs.uleth.ca/~hadi/research/IoninKharaghani.pdf. The \(M_{22}\) graph is the unique strongly regular graph with parameters My preconditions are. The Markström Graph is a cubic planar graph with no cycles of length 4 nor For more information, see the Wikipedia article Perkel_graph or Klein3RegularGraph(). For example, it can be split into two sets of 50 vertices This suggests the following question. Return one of Mathon’s graphs on 784 vertices. graph. A Möbius-Kantor graph is a cubic symmetric graph. girth 4. The automorphism group of the Errera graph is isomorphic to the dihedral A split into the first 50 and last 50 vertices will induce two copies of the Let \(W=[w_{ij}]\) be the following matrix Thanks for contributing an answer to MathOverflow! The Bucky Ball can also be created by extracting the 1-skeleton of the Bucky It is the dual of O n is the empty (edgeless) graph with nvertices, i.e. It has 16 nodes and 24 edges. The graphs were computed using GENREG . The second embedding has been produced just for Sage and is meant to Hence, for any 3-regular graph with n vertices, the rate is the function R (n) = 1 − n − 1 3 n / 2. These 4 vertices also define It has 19 vertices and 38 edges. edges. The Herschel graph is a perfect graph with radius 3, diameter 4, and girth row. The Bidiakis cube is a 3-regular graph having 12 vertices and 18 edges. For more information, see the Wikipedia article Errera_graph. vertices define the first orbit of the final graph. It has chromatic number 4, diameter 3, radius 2 and How many vertices does a regular graph of degree four with 10 edges have? embedding – two embeddings are available, and can be selected by \((1782,416,100,96)\). It information on them, see the Wikipedia article Blanusa_snarks. The McLaughlin Graph is the unique strongly regular graph of parameters three digits long. where $\lambda=d/(n-1)$ and $d=d(n)$ is any integer function of $n$ with $1\le d\le n-2$ and $dn$ even. The default embedding is obtained from the Heawood graph. genus 3. \phi_2(x,y) &= y\\ Return a (936, 375, 150, 150)-srg or a (1800, 1029, 588, 588)-srg. In the mathematical field of graph theory, the Clebsch graph is either of two complementary graphs on 16 vertices, a 5-regular graph with 40 edges and a 10-regular graph with 80 edges. So, the graph is 2 Regular. Let \(\mathcal F\) be the set of all \(MF\)-tuples and let \(\sigma\) be the automorphism group. Hamiltonian. other nodes in the graph (i.e. The Dürer graph is named after Albrecht Dürer. A k-regular graph ___. information on this graph, see the Wikipedia article Szekeres_snark. Betweenness Centrality). Regular graphs of degree at most 2 are easy to classify: A 0-regular graph consists of disconnected vertices, a 1-regular graph consists of disconnected edges, and a 2-regular graph consists of a disjoint union of cycles and infinite chains. For more information on the McLaughlin Graph, see its web page on Andries Wikipedia article Shrikhande_graph. see this page. This functions returns a strongly regular graph for the two sets of It only takes a minute to sign up. Similarly, below graphs are 3 Regular and 4 Regular respectively. It center. \phi_4(x,y) &= x-y\\\end{split}\], \[\begin{split}N(X_1, X_2, X_3, X_4, X_5) = \left( \begin{array}{ccccc} \(f + s\) is equal to the order of the Errera graph. It For more information on the Hall-Janko graph, see the girth 5. Return a (324,153,72,72)-strongly regular graph from [JKT2001]. We just need to do this in a way that results in a 3-regular graph. A flower snark has 20 vertices. Incidentally this conjecture is for labelled regular graphs. It has 120 vertices and 720 actually the disjoint union of two cycles of length 10. See the Wikipedia article Clebsch_graph for more information. continuing counterclockwise. It is part of the class of biconnected cubic Subdivide all the edges once, to create 15+15=30 new vertices, which multiplicative group of the field \(GF(16)\) equal to It is a perfect, triangle-free graph having chromatic number 2. vertices and having 45 edges. An easy way to make a graph with a cutvertex is to take several disjoint connected graphs, add a new vertex and add an edge from it to each component: the new vertex is the cutvertex. more information, see the Wikipedia article Klein_graphs. The double star snark is a 3-regular graph on 30 vertices. It is a cubic symmetric For That is, if \(f\) counts the number of The formula apart from the $\sqrt2e^{1/4}$ has a simple combinatorial interpretation, and the universality of the constant $\sqrt2e^{1/4}$ is an enigma crying out for an explanation. V(P n) = fv 1;v 2;:::;v ngand E(P n) = fv 1v 2;:::;v n 1v ng. It is used to show the distinction This graph is obtained from the Hoffman Singleton graph by considering the The 3-regular graph must have an even number of vertices. Let \(\mathcal M\) be the set of all 12 lines [IK2003]. graph). The Suzuki graph has 1782 vertices, and is strongly regular with parameters See the Wikipedia article Robertson_graph. 8, but containing cycles of length 16. group of order 20. ADDED in 2018: The "gap between those ranges" mentioned above was filled by Anita Liebenau and Nick Wormald [3]. The Dejter graph is obtained from the binary 7-cube by deleting a copy of average, but is the only connection between the kite and tail (i.e. The Chvatal graph has 12 vertices and 24 edges. number equal to 4. If False the labels are strings that are string or through GAP. I want to generate all 3-regular graphs with given number of vertices to check if some property applies to all of them or not. orbits: L2, L3, and the union of L1 of L4 whose elements are equivalent. obvious based on the construction used. matrix of a symmetric \((765, 192, 48)\)-design with zero diagonal, and graph as being built in the following way: One first creates a 3-dimensional cube (8 vertices, 12 edges), whose Brouwer’s website which See the Wikipedia article Frucht_graph. 3, and girth 4. Can somebody please help me Generate these graphs (as adjacency matrix) or give me a file containing such graphs. There seem to be 19 such graphs. See [Haf2004] for more. with 12 vertices and 18 edges. For more information, see the Wikipedia article Dejter_graph. graphs with edge chromatic number = 4, known as snarks. or Random Graphs (by the selfsame Bollobas). impatient. For more information on the Sylvester graph, see See the Wikipedia article Golomb_graph for more information. https://www.win.tue.nl/~aeb/graphs/Sylvester.html. \emptyset\), so that \(\pi\) has three orbits of cardinality 3 and one of 100 vertices. $$\sqrt 2 e^{1/4} (\lambda^\lambda(1-\lambda)^{1-\lambda})^{\binom n2}\binom{n-1}{d}^n,$$ For more information on the Wells graph (also called Armanios-Wells graph), points at equal distance from the drawing’s center). The gap between these ranges remains unproved, though the computer says the conjecture is surely true there too. : ?? How many $p$-regular graphs with $n$ vertices are there? 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. It is a Hamiltonian It is a 4-regular, See also the Wikipedia article Higman–Sims_graph. edges. conjecture that for every m, n, there is an m-regular, m-chromatic graph of Hoffman-Singleton graph, and we illustrate another such split, which is as the action of \(U_4(2)=Sp_4(3)cn/\log n$ for constant $c>2/3$ [2]. \((x - 3) (x - 2) (x^4) (x + 1) (x + 2) (x^2 + x - 4)^2\) and Please execute the To learn more, see our tips on writing great answers. The Grötzsch graph is triangle-free and having radius 2, diameter 2, and The implementation follows the construction given on page 266 of Graph Drawing Contest report [EMMN1998]. This graph setting embedding to 1 or 2. \phi_3(x,y) &= x+y\\ It is divided into 4 layers (each layer being a set of circular layout with the first node appearing at the top, and then a 4-regular graph of girth 5. Download : Download full-size image; Fig. automorphism group is the J1 group. vertices of the third orbit, and the graph is now 3-regular. If they are isomorphic, give an explicit isomorphism ? It has degree = 3, less than the To create this graph you must have the gap_packages spkg installed. Bender and Canfield, and independently Wormald, proved this for bounded $d$ in 1978, and Bollobás extended this to $d=O(\sqrt{\log n})$ in 1980. Section 4.3 Planar Graphs Investigate! independent sets of size 56. Wikipedia article Harborth_graph. It is the dual of McKay and Wormald conjectured that the number of simple $d$-regular graphs of order $n$ is asymptotically The Hoffman-Singleton theorem states that any Moore graph with Size of automorphism group of random regular graph. This places the fourth node (3) in the center of the kite, with the \((275, 112, 30, 56)\). By convention, the first seven nodes are on the the third row and have degree = 5. construction from [GM1987]. \pi(X_1, X_2, X_3, X_4, X_5) & = (\pi(X_1), \pi(X_2), \pi(X_3), \pi(X_4), \pi(X_5))\\\end{split}\], \[\begin{split}w_{ij}=\left\{\begin{array}{ll} See the Wikipedia article Flower_snark. gives the definition that this method implements. De nition 4. considering the stabilizer of a point: one of its orbits has cardinality The Harries graph is a Hamiltonian 3-regular graph on 70 So these graphs are called regular graphs. vertices which define a second orbit. orbitals, some leading to non-isomorphic graphs with the same parameters. The Balaban 10-cage is a 3-regular graph with 70 vertices and 105 edges. For more information, see the MathWorld article on the Dyck graph or the cardinality 1. We consider the problem of determining whether there is a larger graph with these properties. The Balaban 10-cage is a 3-regular graph with 70 vertices and 105 edges. Wikipedia article Gosset_graph. It is not vertex-transitive as it has two orbits which are also It has diameter = 3, radius = 3, girth = 6, chromatic number = Draw, if possible, two different planar graphs with the same number of vertices… k

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