A polyhedral 7-vertex graph is a polyhedral graph on seven vertices. There are 34 nonisomorphic
polyhedral 7-vertex graphs. Through duality, they correspond
to the 34 convex heptahedra enumerated by Kirkman
(1862-1863) and Hermes (1899ab, 1900, 1901), as documented by Federico (1969) and
Duijvestijn and Federico (1981). Six are self-dual
graphs, including the wheel graph.
The name "heptahedral graph" is also used for this class (Gayathri 2018, p. 50).
Britton, D. and Dunitz, J. D. "A Complete Catalogue of Polyhedra with Eight or Fewer Vertices." Acta Cryst. A29,
362-371, 1973. https://doi.org/10.1107/S0567739473000963.Duijvestijn,
A. J. W. and Federico, P. J. "The Number of Polyhedral (3-Connected
Planar) Graphs." Math. Comput.37, 523-532, 1981. https://doi.org/10.1090/S0025-5718-1981-0628713-3.Federico,
P. J. "Enumeration of Polyhedra: The Number of 9-Hedra." J. Combin.
Th.7, 155-161, 1969. https://doi.org/10.1016/S0021-9800(69)80050-5.Gayathri,
A. "A Study on Higher Order Triangular Sum Numbers of Polyhedral and Graceful
Graphs." Math. Sci. Internat. Res. J.7, 48-55, 2018. https://kanchiuniv.ac.in/wp-content/uploads/2022/02/Research-Compendium-2018.pdf.Grünbaum,
B. Convex
Polytopes. New York: Wiley, pp. 288 and 424, 1967.Hermes,
O. "Die Formen der Vielflache. II." J. reine angew. Math.120,
305-353, 1899b.Hermes, O. "Die Formen der Vielflache. I."
J. reine angew. Math.120, 27-59, 1899a.Hermes, O. "Die
Formen der Vielflache. III." J. reine angew. Math.122, 124-154,
1900.Hermes, O. "Die Formen der Vielflache. IV." J. reine
angew. Math.123, 312-342, 1901.House of Graphs. Polyhedral
7-Vertex Graphs. Wheel W_7,
and others.Kirkman, T. P. "Application of the Theory of the
Polyhedra to the Enumeration and Registration of Results." Proc. Roy. Soc.
London12, 341-380, 1862-1863.Pegg, E. Jr. "MathPuzzle:
The 34 Convex Heptahedra and Their Characteristic Polynomials." Jun. 1,
2002. https://www.mathpuzzle.com/charpoly.htm.
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