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The cocktail party graph of order , also called the hyperoctahedral graph (Biggs 1993, p. 17) or Roberts graph, is the graph consisting of two rows of paired nodes in which all nodes but the paired ones are connected with a graph edge. It is the graph complement of the ladder rung graph , and the dual graph of the hypercube graph . It is the skeleton of the -cross polytope.This graph arises in the handshake problem. It is a complete n-partite graph that is denoted by Brouwer et al. (1989, pp. 222-223), and is distance-transitive, and hence also distance-regular.The cocktail party graph of order is isomorphic to the circulant graph . The -cocktail party graph is also the -Turán graph.Special cases are summarized in the following table.-cocktail party graph1empty graph 2square graph 3octahedral graph416-cell graphThe -cocktail party graph has independence polynomialwith corresponding recurrence equation..

"The" octahedral graph is the 6-node 12-edge Platonic graph having the connectivity of the octahedron. It is isomorphic to the circulant graph , the cocktail party graph , the complete tripartite graph , and the 4-dipyramidal graph. Several embeddings of this graph are illustrated above.It is implemented in the Wolfram Languageas GraphData["OctahedralGraph"].The octahedral graph has 6 nodes, 12 edges, vertex connectivity 4, edge connectivity 4, graph diameter 2, graph radius 2, and girth 3. It is the unique 6-node quartic graph, and is also a quartic symmetric graph. It has chromatic polynomialand chromatic number 3. It is an integral graph with graph spectrum . Its automorphism group is of order .The octahedral graph is the line graph of the tetrahedralgraph.There are three minimal integral drawings of the octahedral graph, illustrated above, all with maximum edge length of 7 (Harborth and Möller 1994).The..

"The" tetrahedral graph is the Platonic graph that is the unique polyhedral graph on four nodes which is also the complete graph and therefore also the wheel graph . It is implemented in the Wolfram Language as GraphData["TetrahedralGraph"].The tetrahedral graph has a single minimal integral drawing, illustrated above (Harborth and Möller 1994), with maximum edge length 4.The minimal planar integral drawing of the tetrahedral graph, illustrated above, has maximum edge length of 17 (Harborth et al. 1987). The tetrahedral graph is also graceful (Gardner 1983, pp. 158 and 163-164).The tetrahedral graph has 4 nodes, 6 edges, vertex connectivity 4, edge connectivity 3, graph diameter 1, graph radius 1, and girth 3. It has chromatic polynomial(1)(2)and chromatic number 4. It is planarand cubic symmetric.The tetrahedral graph is an integral graph with graph spectrum . Its automorphism group has order .The..

A Möbius ladder, sometimes called a Möbius wheel (Jakobson and Rivin 1999), of order is a simple graph obtained by introducing a twist in a prism graph of order that is isomorphic to the circulant graph . Möbius ladders are sometimes denoted .The 4-Möbius ladder is known as the Wagner graph. The -Möbius ladder rung graph is isomorphic to the Haar graph .Möbius ladders are Hamiltonian. They are also graceful(Gallian 1987, Gallian 2018).The numbers of directed Hamiltonian cycles for , 4, ... are 12, 10, 16, 14, 20, 18, 24, ... (OEIS A124356), given by the closed form(1)The -Möbius ladder graph has independence polynomial(2)Recurrence equations for the independence polynomial and matching polynomial are given by(3)(4)The bipartite double graph of the -Möbius ladder is the prism graph ...

In graph theory, a cycle graph , sometimes simply known as an -cycle (Pemmaraju and Skiena 2003, p. 248), is a graph on nodes containing a single cycle through all nodes. A different sort of cycle graph, here termed a group cycle graph, is a graph which shows cycles of a group as well as the connectivity between the group cycles. Cycle graphs can be generated in the Wolfram Language using CycleGraph[n]. Precomputed properties are available using GraphData["Cycle", n]. A graph may be tested to see if it is a cycle graph using PathGraphQ[g] && Not[AcyclicGraphQ[g]], where the second check is needed since the Wolfram Language believes cycle graphs are also path graphs (a convention which seems nonstandard at best).Special cases include (the triangle graph), (the square graph, also isomorphic to the grid graph ), (isomorphic to the bipartite Kneser graph ), and (isomorphic to the 2-Hadamard graph). The -cycle graph is isomorphic..

A complete graph is a graph in which each pair of graph vertices is connected by an edge. The complete graph with graph vertices is denoted and has (the triangular numbers) undirected edges, where is a binomial coefficient. In older literature, complete graphs are sometimes called universal graphs.The complete graph is also the complete n-partite graph .The complete graph on nodes is implemented in the Wolfram Language as CompleteGraph[n]. Precomputed properties are available using GraphData["Complete", n]. A graph may be tested to see if it is complete in the Wolfram Language using the function CompleteGraphQ[g].The complete graph on 0 nodes is a trivial graph known as the null graph, while the complete graph on 1 node is a trivial graph known as the singleton graph.In the 1890s, Walecki showed that complete graphs admit a Hamilton decomposition for odd , and decompositions into Hamiltonian cycles plus a perfect matching for..

A circulant graph is a graph of graph vertices in which the th graph vertex is adjacent to the th and th graph vertices for each in a list . The circulant graph gives the complete graph and the graph gives the cyclic graph .The circulant graph on vertices on an offset list is implemented in the Wolfram Language as CirculantGraph[n, l]. Precomputed properties are available using GraphData["Circulant", n, l].With the exception of the degenerate case of the path graph , connected circulant graphs are biconnected, bridgeless, cyclic, Hamiltonian, LCF, regular, traceable, and vertex-transitive.A graph is a circulant iff the automorphism group of contains at least one permutation consisting of a minimal cycle of length .The numbers of circulant graphs on , 2, ... nodes (counting empty graphs as circulant graphs) are 1, 2, 2, 4, 3, 8, 4, 12, ... (OEIS A049287), the first few of which are illustrated above. Note that these numbers cannot be counted..

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