Hoffman–Singleton graph

Here are some constructions of the Hoffman–Singleton graph.{{citation|url=http://blogs.ams.org/visualinsight/2016/02/01/hoffman-singleton-graph/|title=Hoffman–Singleton Graph|first=John|last=Baez|authorlink=John Baez|date=February 1, 2016|publisher=American Mathematical Society|work=Visual Insight}}

Take five pentagons P_h and five pentagrams Q_i . Join vertex j of P_h to vertex h · i + j of Q_i (all indices are modulo 5.)

Take a Fano plane on seven elements, such as {abc, ade, afg, bef, bdg, cdf, ceg} and apply all 2520 even permutations on the {{nowrap|7-set}} abcdefg. Canonicalize each such Fano plane (e.g. by reducing to lexicographic order) and discard duplicates. Exactly 15 Fano planes remain. Each {{nowrap|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.

Let $G$ be the set $\backslash mathbb\{Z\}\_2\backslash times\backslash mathbb\{Z\}\_5\backslash times\backslash mathbb\{Z\}\_5$. Define a binary operation $\backslash circ$ on $G$ such that for each $(a,b,c)$ and $(x,y,z)$ in $G$,

$(a,b,c)\backslash circ(x,y,z)=(a+x,b-bx+y,c+(-1)^a\; by+2^az)$.

Then the Hoffman-Singleton graph has vertices $g\; \backslash in\; G$ and that there exists an edge between $g\; \backslash in\; G$ and $g\text{'}\backslash in\; G$ whenever $g\text{'}=g\backslash circ\; s$ for some $s\; \backslash in\; \backslash \{(0,0,1),(0,0,4),(1,0,0),(1,1,0),(1,2,0),(1,3,0),(1,4,0)\backslash \}$.

{{cite arXiv | last1=Chishwashwa | first1=N. | last2=Faber | first2=V.|last3=Streib | first3=N. | title=Digraph Networks and Groupoids | eprint=2208.10537 | year=2022| class=math.CO }} (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.{{citation | last1=Harder | first1=Jannis | title=Messages on Mastodon | url=https://mathstodon.xyz/@jix/111812567568668511}})

The automorphism group of the Hoffman–Singleton graph is a group of order {{formatnum:252000}} 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 recursivelyThese generators published by GAP. There are of course many other generators for this group.

:$(1\backslash \; 44\backslash \; 22\backslash \; 49\backslash \; 17\backslash \; 43\backslash \; 9\backslash \; 46\backslash \; 40\backslash \; 45)$

(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

:$(1\backslash \; 7\backslash \; 48\backslash \; 47\backslash \; 41\backslash \; 46\backslash \; 17)$

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

There are 1260 5-cycles and 5250 6-cycles in the Hoffman–Singleton graph. There are 525 copies of the Petersen graph, with each {{nowrap|6-cycle}} belonging to exactly one Petersen each. Removing any one Petersen leaves a copy of the unique {{nowrap|(6,5)-cage}}. The Hoffman Singleton graph also contains the odd graph O(4), the Coxeter graph, and the Tutte-Coxeter graph as subgraphs.{{citation

| url = http://www.win.tue.nl/~aeb/drg/graphs/Hoffman-Singleton.html

| title = Hoffman-Singleton graph

| authorlink = Andries E. Brouwer

| author = Brouwer, Andries E.}}.

Take any edge of the Hoffman-Singleton graph, and discard those two vertices as well as the 12 vertices directly connected to either of them. The graph that remains is the Sylvester graph on 36 vertices. Because each such edge can be mapped to a distinct Sylvester graph, there are 175 copies of the Sylvester graph in the Hoffman Singleton graph.

The Hoffman Singleton graph is contained in the Higman–Sims graph which is therefore a supergraph.

• McKay–Miller–Širáň graphs, a class of graphs including the Hoffman–Singleton graph
• Table of the largest known graphs of a given diameter and maximal degree

{{reflist}}

• {{Citation | last1=Benson | first1=C. T. | last2=Losey | first2=N. E. | title=On a graph of Hoffman and Singleton | doi=10.1016/0095-8956(71)90015-3 |mr=0281658 | year=1971 | journal=Journal of Combinatorial Theory, Series B | issn=0095-8956 | volume=11 | issue=1 | pages=67–79| doi-access= }}
• {{cite arXiv | last1=Chishwashwa | first1=N. | last2=Faber | first2=V.|last3=Streib | first3=N. | title=Digraph Networks and Groupoids | eprint=2208.10537 | year=2022 | class=math.CO |mode=cs2}}
• {{citation

| last1 = Holton | first1 = D.A.

| last2 = Sheehan | first2 = J.

| isbn = 0-521-43594-3

| pages = 186 and 190

| publisher = Cambridge University Press

| title = The Petersen graph

| title-link = The Petersen Graph

| year = 1993}}.

{{DEFAULTSORT:Hoffman-Singleton Graph}}

Category:Individual graphs

Category:Regular graphs

Category:Strongly regular graphs