Hoffman–Singleton graph

Hoffman–Singleton graph

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The two circles shaped Hoffman–Singleton graph and its component (50 + 25 + 25 + 25 + 25 + 25 = 175 edges).

The Hoffman–Singleton graph is a graph with the following properties:
* The graph has 50 vertices.
* The graph has 175 edges.
* The graph has a vertex degree of 7.
* The graph has a diameter of 2.
* The graph has a girth of 5.

Therefore, the graph is the following:
* strongly regular.
* A Moore graph.
* An integral graph.
* A symmetric graph.
* The (7,5)-cage.

A Hoffman–Singleton graph is the highest order Moore graph known to exist.All Hoffman–Singleton graphs will adhere to all eight properties listed above—no matter how they are drawn.

References

* D.A. Holton and J. Sheehan, "The Petersen graph", Cambridge University Press, 1993, ISBN 0-521-43594-3. Pp.186,190.
* citation
last1 = Hoffman
first1 = Alan J.
last2 = Singleton
first2 = Robert R.
title = Moore graphs with diameter 2 and 3
journal = IBM Journal of Research and Development
volume = 5
issue = 4
year = 1960
pages = 497–504
url = http://www.research.ibm.com/journal/rd/045/ibmrd0405H.pdf
id = MathSciNet | id = 0140437


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