Английская Википедия:Hoffman graph
In the mathematical field of graph theory, the Hoffman graph is a 4-regular graph with 16 vertices and 32 edges discovered by Alan Hoffman.[1] Published in 1963, it is cospectral to the hypercube graph Q4.[2][3]
The Hoffman graph has many common properties with the hypercube Q4—both are Hamiltonian and have chromatic number 2, chromatic index 4, girth 4 and diameter 4. It is also a 4-vertex-connected graph and a 4-edge-connected graph. However, it is not distance-regular. It has book thickness 3 and queue number 2.[4]
Algebraic properties
The Hoffman graph is not a vertex-transitive graph and its full automorphism group is a group of order 48 isomorphic to the direct product of the symmetric group S4 and the cyclic group Z/2Z.
The characteristic polynomial of the Hoffman graph is equal to
- <math>(x-4) (x-2)^4 x^6 (x+2)^4 (x+4)</math>
making it an integral graph—a graph whose spectrum consists entirely of integers. It is the same spectrum as the hypercube Q4.
Gallery
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The Hoffman graph is Hamiltonian.
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The chromatic number of the Hoffman graph is 2.
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The chromatic index of the Hoffman graph is 4.
References
- ↑ Шаблон:MathWorld
- ↑ Hoffman, A. J. "On the Polynomial of a Graph." Amer. Math. Monthly 70, 30-36, 1963.
- ↑ van Dam, E. R. and Haemers, W. H. "Spectral Characterizations of Some Distance-Regular Graphs." J. Algebraic Combin. 15, 189-202, 2003.
- ↑ Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018
