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616-CellThe 16-cell is the finite regular four-dimensional. It is also known as the hyperoctahedron or hexadecachoron, and its composed of 16 tetrahedra, with 4 to an edge. It has 8 vertices, 24 edges, and 32 faces. The 16-cell. It has distinct nets.
724-CellThe 24-cell is a finite regular four-dimensional. It is also known as the hyperdiamond or icositetrachoron, and is composed of 24 octahedra, with 3 to an edge. The 24-cell has 24 vertices and 96 edges. It is one of the six regular polychora. The 24-cell hasdistinct nets.
8120-CellThe 120-cell is a finite regular four-dimensional, also known as the hyperdodecahedron or hecatonicosachoron, and composed of 120 dodecahedra, with 3 to an edge, and 720 pentagons.The 120-cell has 600 vertices and 1200 edges.
10600-CellThe 600-cell is the finite regular four-dimensional. It is also known as the hypericosahedron or hexacosichoron. It is composed of 600 tetrahedra, with 5 to an edge. The 600-cell has 120 vertices and 720 edges.
11Balaban 10-CageIt is a Hamiltonian graph and has Hamiltonian cycles. The Balaban 10-cage is one of the three -cage graphs.
12Bidiakis CubeThe 12-vertex graph consisting of a cube in which two opposite faces (say, top and bottom) have edges drawn across them which connect the centers of opposite sides of the faces in such a way that the orientation of the edges added on top and bottom are perpendicular to each other.
13Biggs-Smith graphThe Biggs-Smith graph is cubic symmetric graph on 102 vertices and 153 edges that is also distance-regular.
14Bislit CubeThe bislit cube is the 8-vertex simple graph consisting of a cube in which two opposite faces have polyhedron diagonals oriented perpendicular to each other.
15Brinkmann GraphThe Brinkmann graph is a weakly regular quartic graph on 21 vertices and 42 edges.
16Clebsch GraphThe Clebsch graph, also known as the Greenwood-Gleason graph , is a strongly regular quintic graph on 16 vertices and 40 edges
17Cubical GraphThe cubical graph is the Platonic graph corresponding to the connectivity of the cube. It is equivalent to the generalized Petersen graph. The cubical graph has 8 nodes, 12 edges.
18Cuboctahedral GraphAn Archimedean symmetric quartic graph on 12 nodes and 24 edges that is the skeleton of the cuboctahedron. The cuboctahedral graph is the line graph of the cubical graph.
19Desargues GraphThe Desargues graph is a cubic symmetric graph distance-regular graph on 20 vertices and 30 edges.
20Diamond GraphThe diamond graph is the simple graph on 4 nodes and 5 edges illustrated above.
21Disdyakis Dodecahedral Graph The disdyakis dodecahedral graph is Archimedean dual graph which is the skeleton of the disdyakis dodecahedron.
23Dodecahedral GraphThe dodecahedral graph is the Platonic graph corresponding to the connectivity of the vertices of a dodecahedron. The dodecahedral graph has 20 nodes, 30 edges.
24Dyck GraphThe unique cubic symmetric graph on 32 nodes 48 edges. It is nonplanar.
25Errera GraphThe Errera graph is the 17-node planar graph. It is an example of how Kempe's supposed proof of the four-color theorem fails.
26Folkman GraphThe Folkman graph is a semisymmetric graph that has the minimum possible number of nodes (20)
27Foster GraphThe" Foster graph is the cubic symmetric graph on 90 vertices that has 135 edges
28Wong GraphThe Wong graph is one of the four (5,5) -cage graphs. Like the other (5,5) -cages, the Wong graph has 30 nodes. It has 75 edges, girth 5, diameter 3, chromatic number 4, and is a quintic graph.
29Wells GraphThe Wells graph is a quintic graph on 32 nodes and 80 edges that is the unique distance-regular graph with intersection array (5, 4, 1, 1; 1, 1, 4, 5) . It is a double cover of the complement of the Clebsch graph (Brouwer et al. 1989, p. 266).
30Utility GraphThe utility problem posits three houses and three utility companies--say, gas, electric, and water--and asks if each utility can be connected to each house without having any of the gas/water/electric lines/pipes pass over any other. This is equivalent to the equation "Can a planar graph be constructed from each of three nodes ('houses') to each of three other nodes ('utilities')?" This problem was first posed in this form by H. E. Dudeney in 1917 (Gardner 1984, p. 92).
31Unitransitive GraphA graph G is n -unitransitive if it is connected, cubic, n-transitive, and if for any two n-routes W1 and W2
32Truncated Tetrahedral Graph The truncated tetrahedral graph is the cubic Archimedean graph on 12 nodes and 18 edges that is the skeleton of the truncated tetrahedron.
33Truncated Octahedral Graph The truncated octahedron graph is the cubic Archimedean graph on 24 nodes and 36 edges that is the skeleton of the truncated octahedron.
34Truncated Icosahedral Graph The truncated icosahedral graph is the cubic Archimedean graph on 60 nodes and 90 edges that is the skeleton of the truncated icosahedron. A number of embeddings are shown above.
44Snub Dodecahedral Graph The snub dodecahedral graph is a quintic graph on 60 nodes and 150 edges that corresponds to the skeleton of the snub dodecahedron. The snub dodecahedral graph is planar and Hamiltonian, and has chromatic number 4. It is vertex-transitive, although not edge-transitive because some edges are part of three-circuits while others are not.
45Snub Cubical GraphThe snub cubical graph is the Archimedean graph on 24 nodes and 60 edges obtained by taking the skeleton of the snub cube. It is a quintic graph, is planar, Hamiltonian, and has chromatic number 3.
46Small Rhombicuboctahedral Graph The small rhombicuboctahedral graph is a quartic graph on 24 nodes and 48 edges that corresponds to the skeleton of the small rhombicuboctahedron. It has graph diameter 5, graph radius 5, and chromatic number 3. It is also Hamiltonian. It is vertex-transitive, although not edge-transitive because some edges are part of three-circuits while others are not.
47Small Rhombicosidodecahedral Graph The small rhombicosidodecahedral graph is a quartic graph on 60 nodes and 120 edges that corresponds to the skeleton of the small rhombicosidodecahedron. It has graph diameter 8, graph radius 8, and chromatic number 3. It is also Hamiltonian. It is vertex-transitive, although not edge-transitive because some edges are part of three-circuits while others are not.
48Robertson-Wegner Graph The Robertson-Wegner graph has 30 nodes. It has 75 edges, girth 5, diameter 3, and chromatic number 4.
49Robertson GraphThe Robertson graph has 19 vertices and 38 edges.
50PentatopeThe pentatope is self-dual, has five three-dimensional facets (each the shape of a tetrahedron), 10 ridges (faces), 10 edges, and five vertices.
51Pentakis Dodecahedral Graph The pentakis dodecahedral graph is Archimedean dual graph which is the skeleton of the disdyakis triacontahedron.
52Pentagonal Icositetrahedral Graph The pentagonal icositetrahedral graph is the Archimedean dual graph which is the skeleton of the pentagonal icositetrahedron.
53Pentagonal Hexecontahedral Graph The pentagonal hexecontahedral graph is the Archimedean dual graph which is the skeleton of the pentagonal hexecontahedron.
54Pappus GraphA cubic symmetric distance-regular graph on 18 vertices, illustrated above in three embeddings.
55Octahedral GraphThe 6-node 12-edge Platonic graph having the connectivity of the octahedron
56Möbius-Kantor GraphThe unique cubic symmetric graph on 16 nodes, illustrated above in two embeddings.
57Meringer GraphThe Meringer graph has 30 nodes. It has 75 edges, girth 5, diameter 3, chromatic number 3, and is a quintic graph. The order of its automorphism group is 96.
58McGee GraphThe McGee graph was discovered by McGee (1960) and proven unique by Tutte (1966; Wong 1982). It has 24 nodes, 36 edges, girth 7, diameter 4, and is a cubic graph.
59Levi Graph"The" Levi graph (right figure) is a graph based on Desargues' configuration which consists of the union of the two leftmost subgraphs illustrated above. It has 30 nodes and 45 edges. It has girth 8, diameter 4, chromatic number 2, and automorphism group order 1440.
61Kittell GraphThe Kittell graph is a planar graph on 23 nodes and 63 edges that tangles the Kempe chains in Kempe's algorithm and thus provides an example of how Kempe's supposed proof of the four-color theorem fails.
62Icosidodecahedral Graph A symmetric quartic graph on 30 nodes and 60 edges corresponding to the skeleton of the icosidodecahedron. It has graph diameter 5, graph radius 5, and chromatic number 3. It is also Hamiltonian.
63Icosahedral GraphThe icosahedral graph is the Platonic graph whose nodes have the connectivity of the icosahedron, illustrated above in a number of embeddings. The icosahedral graph has 12 vertices and 30 edges.
71Harries-Wong GraphThe Harries-Wong graph is one of the three (10,3) -cage graphs, the other two being the (10,3) -cage Balaban graph and the Harries graph.
72Harries GraphThe Harries graph is one of the three (10,3) -cage graphs, the other two being the (10,3) -cage Balaban graph and the Harries-Wong graph.
73Harborth GraphThe Harborth graph is the smallest known 4-regular matchstick graph. It is therefore both planar and unit-distance. It has 104 edges and 52 vertices. This graph was named after its discoverer H. Harborth, who first presented it to a general public in 1986 (Harborth 1986, Petersen 1996, Gerbracht 2006).
74Grünbaum GraphThe Grünbaum graph can be constructed from the dodecahedral graph by adding an additional ring of five vertices around the perimeter and cyclically connecting each new vertex to three others as shown above (left figure). A more symmetrical embedding is shown in the right figure above. The Grünbaum graph has 25 vertices and 50 edges.
75Grötzsch GraphThe Grötzsch graph is smallest triangle-free graph with chromatic number four. It is identical to the Mycielski Graph of order four. It has 11 vertices and 20 edges. It is Hamiltonian, but nonplanar.
76Great Rhombicuboctahedral Graph The cubic Archimedean graph on 48 nodes and 72 edges that is the skeleton of the great rhombicuboctahedron.
78Great Rhombicosidodecahedral Graph The great rhombicosidodecahedral graph is the Archimedean graph on 120 vertices and 180 edges that is the skeleton of the great rhombicosidodecahedron. It is cubic, has chromatic number 2, and is planar and Hamiltonian.
79Gray GraphThe Gray graph is a semisymmetric cubic graph on 54 vertices. It was discovered by Marion C. Gray in 1932, and was first published by Bouwer (1968). Malnic et al. (2004) showed that the Gray graph is indeed the smallest possible semisymmetric cubic graph.
80Gewirtz GraphThe Gewirtz graph, sometimes also called the Sims-Gewirtz graph (Brouwer), is an integral graph on 56 nodes and 280 edges that is also a regular graph of order 10.
81Frucht GraphThe Frucht graph is smallest cubic identity graph (Skiena 1990, p. 185). It has 12 vertices and 18 edges. It is also both planar and Hamiltonian.
82ทฤษฎีบทของดิแรก(Dirac’s theoem) กราฟเชิงเดียวที่มีจุดยอด n จุด เมื่อ n >= 3 และจุดยอดทุกจุดมีดีกรีอย่างน้อย n/2 แล้วกราฟดังกล่าวเป็นกราฟแฮมิลตัน