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.^[2] Published in 1963, it is cospectral to the hypercube graph Q₄.^[3][4]

Hoffman graph
The Hoffman graph
Named after	Alan Hoffman
Vertices	16
Edges	32
Radius	3
Diameter	4
Girth	4
Automorphisms	48 (Z/2Z × S4)
Chromatic number	2
Chromatic index	4
Book thickness	3
Queue number	2
Properties	Hamiltonian[1] Bipartite Perfect Eulerian 1-walk regular
Table of graphs and parameters

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.^[5]

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

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

making it an integral graph—a graph whose spectrum consists entirely of integers. It is the same spectrum as the hypercube Q₄.

Gallery

• 
  The Hoffman graph is Hamiltonian.
• 
  The chromatic number of the Hoffman graph is 2.
• 
  The chromatic index of the Hoffman graph is 4.