Through duality, these graphs correspond to the 257 convex octahedra with eight
faces enumerated by Kirkman (1862-1863) and Hermes (1899ab,
1900, 1901), as documented by Federico (1969) and Duijvestijn and Federico (1981).
Britton and Dunitz (1973) catalogue convex polyhedra
by vertex count up to eight vertices.
The name "octahedral graph" is also used for this class (Gayathri 2018, p. 50). It is distinguished here from the six-vertexoctahedral graph, the skeleton
of the regular octahedron.
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. I." J. reine angew. Math.120,
27-59, 1899a.Hermes, O. "Die Formen der Vielflache. II." J.
reine angew. Math.120, 305-353, 1899b.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
8-Vertex Graphs. The cube Q_3,
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.
