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Hypercube Graph

The -hypercube graph, also called the -cube graph and commonly denoted or , is the graph whose vertices are the symbols , ..., where or 1 and two vertices are adjacent iff the symbols differ in exactly one coordinate.The graph of the -hypercube is given by the graph Cartesian product of path graphs . The -hypercube graph is also isomorphic to the Hasse diagram for the Boolean algebra on elements.The above figures show orthographic projections of some small -hypercube graphs using the first two of each vertex's set of coordinates. Note that above is a projection of the usual cube looking along a space diagonal so that the top and bottom vertices coincide, and hence only seven of the cube's eight vertices are visible. In addition, three of the central edges connect to the upper vertex, while the other three connect to the lower vertex.Hypercube graphs may be computed in the Wolfram Language using the command HypercubeGraph[n], and precomputed properties..

Heawood Graph

The Heawood graph is a cubic graph on 14 vertices and 21 edges which is the unique (3,6)-cage graph. It is also a Moore graph. The Heawood graph is also the generalized hexagon , and its line graph is the generalized hexagon . The Heawood graph is illustrated above in a number of embeddings.It has graph diameter 3, graph radius 3, and girth 6. It is cubic symmetric, nonplanar, Hamiltonian, and can be represented in LCF notation as .It has chromatic number 2 and chromaticpolynomialIts graph spectrum is .It is 4-transitive, but not 5-transitive (Harary 1994, p. 173).The Heawood graph is one of eight cubic graphs on 14 nodes with smallest possible graph crossing number of 3 (another being the generalized Petersen graph ), making it a smallest cubic crossing number graph (Pegg and Exoo 2009, Clancy et al. 2019).The Heawood graph corresponds to the seven-color torus map on 14 nodes illustrated above. The Heawood graph is the point/line incidence..

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