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.

[7] Take five pentagons Ph and five pentagrams Qi .

)[7] Take a Fano plane on seven elements, such as {abc, ade, afg, bef, bdg, cdf, ceg} and apply all 2520 even permutations on the 7-set abcdefg.

Canonicalize each such Fano plane (e.g. by reducing to lexicographic order) and discard duplicates.

Each 3-set (triplet) of the set abcdefg is present in exactly 3 Fano planes.

The incidence between the 35 triplets and 15 Fano planes induces PG(3,2), with 15 points and 35 lines.

To make the Hoffman-Singleton graph, create a graph vertex for each of the 15 Fano planes and 35 triplets.

Connect each Fano plane to its 7 triplets, like a Levi graph, and also connect disjoint triplets to each other like the odd graph O(4).

[8] (Although the authors use the word "groupoid", it is in the sense of a binary function or magma, not in the category-theoretic sense.

in the last term, but that does not produce the Hoffman-Singleton graph.

[9]) 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.

As a permutation group on 50 symbols, it can be generated by the following two permutations applied recursively[10] and 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

The Hoffman-Singleton graph has exactly 100 independent sets of size 15 each.

The 100-vertex graph that connects disjoint independent sets can be partitioned into two 50-vertex subgraphs, each of which is isomorphic to the Hoffman-Singleton graph, in an unusual case of self-replicating + multiplying behavior.

Removing any one Petersen leaves a copy of the unique (6,5)-cage.