Hoffman–Singleton graph



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
Here are some constructions of the Hoffman–Singleton graph.

Construction from pentagons and pentagrams
Take five pentagons Ph and five pentagrams Qi. Join vertex j of Ph to vertex h·i+j of Qi. (All indices are modulo 5.)

Construction from PG(3,2)
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. Exactly 15 Fano planes remain. 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).

A very similar construction from PG(3,2) is used to build the Higman–Sims graph, which has the Hoffman-Singleton graph as a subgraph.

Construction on a groupoid/magma
Let $$G$$ be the set $$\mathbb{Z}_2\times \mathbb{Z}_5\times\mathbb{Z}_5$$. Define a binary operation $$\circ$$ on $$G$$ such that for each $$(a,b,c)$$ and $$(x,y,z)$$ in $$G$$, $$(a,b,c)\circ(x,y,z)=(a+x,b-bx+y,c+(-1)^a by+2^az)$$. Then the Hoffman-Singleton graph has vertices $$g \in G$$ and that there exists an edge between $$g \in G$$ and $$g'\in G$$ whenever $$g'=g\circ s$$ for some $$s \in \{(0,0,1),(0,0,4),(1,0,0),(1,1,0),(1,2,0),(1,3,0),(1,4,0)\}$$. (Although the authors use the word "groupoid", it is in the sense of a binary function or magma, not in the category-theoretic sense. Also note there is a typo in the formula in the paper: the paper has $$(-1)^x by$$ in the last term, but that does not produce the Hoffman-Singleton graph. It should instead be $$(-1)^a by $$ as written here. )

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. As a permutation group on 50 symbols, it can be generated by the following two permutations applied recursively $$ (1\ 44\ 22\ 49\ 17\ 43\ 9\ 46\ 40\ 45)

(2\ 23\ 24\ 14\ 18\ 10\ 12\ 42\ 38\ 6)

(3\ 41\ 19\ 4\ 15\ 20\ 7\ 13\ 37\ 8)

(5\ 28\ 21\ 29\ 16\ 25\ 11\ 26\ 39\ 30)

(27\ 47)

(31\ 36\ 34\ 32\ 35)

(33\ 50) $$

and

$$ (1\ 7\ 48\ 47\ 41\ 46\ 17)

(2\ 39\ 11\ 4\ 15\ 14\ 42)

(3\ 32\ 28\ 9\ 23\ 6\ 43)

(5\ 22\ 38\ 18\ 44\ 36\ 29)

(8\ 37\ 40\ 34\ 26\ 49\ 24)

(10\ 16\ 31\ 27\ 13\ 21\ 45)

(19\ 33\ 25\ 35\ 50\ 30\ 20) $$

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.

The Hoffman-Singleton graph has exactly 100 independent sets of size 15 each. Each independent set is disjoint from exactly 7 other independent sets. 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.