List of solution strategies for differential equations

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). 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. Since it is a Moore graph where each vertex has degree 7, and the girth is 5, it is a (7,5)-cage.

Construction

A simple direct construction is the following: Take five pentagons P_h and five pentagrams Q_i, so that vertex j of P_h is adjacent to vertices j-1,j+1 of P_h and vertex j of Q_i is adjacent to vertices j-2,j+2 of Q_i. Now join vertex j of P_h to vertex hi+j of Q_i. (All indices mod 5.)

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 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.

See also

• Table of the largest known graphs of a given diameter and maximal degree

Notes

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References

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• Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.
  
  Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.
  
  In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.
  
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