# Hoffman graph

Hoffman graph
The Hoffman graph
Named after Alan Hoffman
Vertices 16
Edges 32
Diameter 4
Girth 4
Automorphisms 48 (Z/2Z × S4)
Chromatic number 2
Chromatic index 4
Properties Hamiltonian[1]
Bipartite
Perfect
Eulerian

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]

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.

## 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

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

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

## References

1. ^
2. ^
3. ^ Hoffman, A. J. "On the Polynomial of a Graph." Amer. Math. Monthly 70, 30-36, 1963.
4. ^ van Dam, E. R. and Haemers, W. H. "Spectral Characterizations of Some Distance-Regular Graphs." J. Algebraic Combin. 15, 189-202, 2003.