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 Q<sub>4</sub>.
The Hoffman graph has many common properties with the hypercube Q<sub>4</sub>âÂÂboth 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 and not 1-planar. It has book thickness 3 and queue number 2.
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 S<sub>4</sub> 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
making it an integral graphâÂÂa graph whose spectrum consists entirely of integers. It is the same spectrum as the hypercube Q<sub>4</sub>.