Hoffman–Singleton graph

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Hoffman–Singleton graph
Named after Alan J. Hoffman
Robert R. Singleton
Vertices 50
Edges 175
Diameter 2[1]
Girth 5[1]
Automorphisms 252,000
(PSU(3,52):2)[2]
Chromatic number 4
Chromatic index 7[3]
Genus 29[4]
Properties Strongly regular
Symmetric
Hamiltonian
Integral
Cage
Moore graph
The Hoffman–Singleton graph. The subgraph of blue edges is a sum of ten disjoint pentagons.

In the mathematical field of graph theory, the Hoffman–Singleton graph is a 7-regular undirected graph with 50 vertices and 175 edges. It is the unique strongly regular graph with parameters (50,7,0,1).[5] It was constructed by Alan Hoffman and Robert Singleton while trying to classify all Moore graphs, and is the highest order Moore graph known to exist.[6] Since it is a Moore graph where each vertex has degree 7, and the girth is 5, it is a (7,5)-cage.

Construction

There are many ways to construct the Hoffman-Singleton graph.

Construction from pentagons and pentagrams

Take five pentagons Ph and five pentagrams Qi, so that vertex j of Ph is adjacent to vertices j-1 and j+1 of Ph and vertex j of Qi is adjacent to vertices j-2 and j+2 of Qi. Now join vertex j of Ph to vertex hi+j of Qi. (All indices mod 5.)

Construction from triads and Fano planes

Take the Fano plane and permute its 7 points to make a set of 30 Fanos. Pick any one of these 30 Fanos; there will be another 14 Fanos that share exactly one triplet ("line") with the first one. Take those 15 Fanos and discard the other 15. Take the 7C3 = 35 triads on 7 numbers. Now connect a triad to a Fano that includes it, and connect disjoint triads to each other. The resulting graph is the Hoffman-Singleton graph, with the 50 vertices corresponding to the 35 triads + 15 Fanos, and each vertex has degree 7. Vertices corresponding to Fanos are linked to 7 triads by definition, as the Fano plane has 7 lines. Each triad is linked to 3 different Fanos that include it, and to 4 other triads with which it is disjoint.

Algebraic properties

The automorphism group of the Hoffman–Singleton graph is a group of order 252,000 isomorphic to PΣU(3,52) the semidirect product of the projective special unitary group PSU(3,52) with the cyclic group of order 2 generated by the Frobenius automorphism. It acts transitively on the vertices, on the edges and on the arcs of the graph. Therefore the Hoffman–Singleton graph is a symmetric graph. The stabilizer of a vertex of the graph is isomorphic to the symmetric group S7 on 7 letters. The setwise stabilizer of an edge is isomorphic to Aut(A6)=A6.22, where A6 is the alternating group on 6 letters. Both of the two types of stabilizers are maximal subgroups of the whole automorphism group of the Hoffman-Singleton graph.

The characteristic polynomial of the Hoffman–Singleton graph is equal to $(x-7) (x-2)^{28} (x+3)^{21}$. Therefore the Hoffman–Singleton graph is an integral graph: its spectrum consists entirely of integers.

Subgraphs

Using only the fact that the Hoffman–Singleton graph is a strongly regular graph with parameters (50,7,0,1), it can be shown that there are 1260 5-cycles contained in the Hoffman–Singleton graph.

Additionally, the Hoffman–Singleton graph contains 525 copies of the Petersen graph.