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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]].
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==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 05:25, 5 March 2014


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