Hoffman graph explained

Hoffman graph
Vertices:	16
Edges:	32
Automorphisms:	48 (Z/2Z × S4)
Girth:	4
Diameter:	4
Radius:	3
Chromatic Number:	2
Chromatic Index:	4
Properties:	Hamiltonian Bipartite Perfect Eulerian
Book Thickness:	3
Queue Number:	2

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.^{[1] [2]}

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.^[3]

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

(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.

Notes and References

1. Hoffman, A. J. "On the Polynomial of a Graph." Amer. Math. Monthly 70, 30-36, 1963.
2. van Dam, E. R. and Haemers, W. H. "Spectral Characterizations of Some Distance-Regular Graphs." J. Algebraic Combin. 15, 189-202, 2003.
3. Jessica Wolz, Engineering Linear Layouts with SAT. Master Thesis, University of Tübingen, 2018