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{{infobox graph
| name = Hoffman graph
| image = [[Image:Hoffman graph.svg|220px]]
| image_caption = The Hoffman graph
| namesake = [[Alan Hoffman (mathematician)|Alan Hoffman]]
| vertices = 16
| edges = 32
| automorphisms = 48 ('''Z'''/2'''Z''' × S<sub>4</sub>)
| girth = 4
| diameter = 4
| radius = 3
| chromatic_number = 2
| chromatic_index = 4
| properties = [[Hamiltonian graph|Hamiltonian]]<ref>{{MathWorld|urlname=HamiltonianGraph|title=Hamiltonian Graph}}</ref><br>[[Bipartite graph|Bipartite]]<br>[[Perfect graph|Perfect]]<br>[[Eulerian graph|Eulerian]]
}}
 
In the [[mathematics|mathematical]] field of [[graph theory]], the '''Hoffman graph''' is a 4-[[regular graph]] with 16 vertices and 32 edges discovered by [[Alan Hoffman (mathematician)|Alan Hoffman]].<ref>{{MathWorld|urlname=HoffmanGraph|title=Hoffman graph}}</ref> Published in 1963, it is cospectral to the [[hypercube graph]] Q<sub>4</sub>.<ref>Hoffman, A. J. "On the Polynomial of a Graph." Amer. Math. Monthly 70, 30-36, 1963.</ref><ref>van Dam, E. R. and Haemers, W. H. "Spectral Characterizations of Some Distance-Regular Graphs." J. Algebraic Combin. 15, 189-202, 2003.</ref>
 
The Hoffman graph has many common properties with the hypercube Q<sub>4</sub>—both are [[Hamiltonian graph| Hamiltonian]] and have [[chromatic number]] 2, [[chromatic index]] 4, radius 3, girth 4 and diameter 4. It is also a 4-[[k-vertex-connected graph|vertex-connected graph]] and a 4-[[k-edge-connected graph|edge-connected graph]]. However, it is not [[Distance-regular graph|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 groups|direct product]] of the [[symmetric group]] S<sub>4</sub> and the [[cyclic group]] '''Z'''/2'''Z'''.
 
The [[characteristic polynomial]] of the Hoffman graph is equal to
:<math>(x-4) (x-2)^4 x^6 (x+2)^4 (x+4)</math>
making it an [[integral graph]]—a graph whose [[Spectral graph theory|spectrum]] consists entirely of integers. It is the same spectrum than the hypercube Q<sub>4</sub>.
 
==Gallery==
<gallery>
Image:Hoffman graph hamiltonian.svg|The Hoffman graph is [[Hamiltonian graph| Hamiltonian]].
Image:Hoffman graph 2COL.svg|The [[chromatic number]] of the Hoffman graph is&nbsp;2.
Image:Hoffman graph 4color edge.svg|The [[chromatic index]] of the Hoffman graph is&nbsp;4.
</gallery>
 
== References ==
{{reflist}}
 
[[Category:Individual graphs]]
[[Category:Regular graphs]]

Revision as of 11:58, 23 January 2014

Template:Infobox 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.[1] Published in 1963, it is cospectral to the hypercube graph Q4.[2][3]

The Hoffman graph has many common properties with the hypercube Q4—both are Hamiltonian and have chromatic number 2, chromatic index 4, radius 3, 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

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

Gallery

References

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