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. Published in 1963, it is cospectral to the hypercube graph Q₄.

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.

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. Despite not being vertex- or edge-transitive, the Hoffmann graph is still 1-walk-regular (but not distance-regular).

The characteristic polynomial of the Hoffman graph is equal to

$(x-4)(x-2{)}^4{x}^6(x+2{)}^4(x+4)$

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.