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Truncated Octahedral Graph

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TruncatedOctahedralGraph

The truncated octahedral graph is the cubic Archimedean graph on 24 nodes and 36 edges that is the skeleton of the truncated octahedron. It is isomorphic to the Bruhat graph B_4, meaning it is the graph having all permutations of {1,2,3,4} as vertices with an edge between pairs of permutations that differ by an adjacent transposition.

It is implemented in the Wolfram Language as GraphData["TruncatedOctahedralGraph"].

The truncated octahedral graph is vertex-transitive but not edge-transitive. It is Hamiltonian. It has LCF notations [3,-7,7,-3]^6, [5,-11,11,7,5,-5,-7,-11,11,-5,-7,7]^2, and [-11, 5, -3, -7, -9, 3, -5, 5, -3, 9, 7, 3, -5, 11, -3, 7, 5, -7, -9, 9, 7, -5, -7, 3].

It has graph spectrum Spec(G)=(+/-3)^1(+/-1)^3(+/-1+/-sqrt(2))^3(+/-sqrt(3))^2. Its automorphism group is of group order |Aut(G)|=48.

The truncated octahedral graph is also its own distance-5 graph distance graph.

Its domination number is 8 and its bondage number is 5 (Srivastava 2026). Since the truncated octahedral graph is cubic and planar, its maximum vertex degree is 3, making it a counterexample to the planar bondage conjecture b(G)<=Delta(G)+1 proposed by Dunbar et al. (1998).

See also

Archimedean Graph, Bruhat Graph, Planar Bondage Conjecture, Truncated Octahedron

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References

Dunbar, J. E.; Haynes, T. W.; Teschner, U.; and Volkmann, L. "Bondage, Insensitivity, and Reinforcement." In Domination in Graphs--Advanced Topics (Ed. T. W. Haynes, S. T. Hedetniemi, and P. J. Slater). New York: Marcel Dekker, pp. 471-489, 1998.House of Graphs. "Truncated Octahedral Graph." https://houseofgraphs.org/graphs/1391.Read, R. C. and Wilson, R. J. An Atlas of Graphs. Oxford, England: Oxford University Press, p. 267, 1998.Srivastava, P. R. "The Truncated Octahedral Graph Has Bondage Number Five." 2 Sep 2026. https://arxiv.org/abs/2609.02477.

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Truncated Octahedral Graph

Cite this as:

Weisstein, Eric W. "Truncated Octahedral Graph." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TruncatedOctahedralGraph.html

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