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# Hexahedral Graph

A hexahedral graph is a polyhedral graph on six vertices. There are seven distinct hexahedral graphs (illustrated above) which, through duality, correspond to seven convex hexahedra. The hexahedral graphs were first enumerated by Steiner (1828; Duijvestijn and Federico 1981).

Three of the hexahedral graphs correspond to the skeletons of the pentagonal pyramid (i.e., the wheel graph ), triangular prism, and octahedron (square dipyramid, triangular antiprism). An additional hexahedron can be obtained by truncation of two of the four apexes of the tetrahedron, producing a solid composed of 2 triangles, 2 quadrilaterals, and two pentagons which, like the cube, has 6 faces, 8 vertices, 12 edges and cubic connectivity.

Hexahedron, Polyhedral Graph

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## References

Duijvestijn, A. J. W. and Federico, P. J. "The Number of Polyhedral (3-Connected Planar) Graphs." Math. Comput. 37, 523-532, 1981.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, 1994.Steiner, J. "Problème de situation." Ann. de Math 19, 36, 1828. Reprinted in Jacob Steiner's gesammelte Werke, Band I. Bronx, NY: Chelsea, p. 227, 1971.

Hexahedral Graph

## Cite this as:

Weisstein, Eric W. "Hexahedral Graph." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HexahedralGraph.html