Hoffman–Singleton graph

Key takeaways

Hoffman–Singleton graph belongs to science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hoffman–Singleton graph to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hoffman–Singleton graph from memory before moving on to harder problems.

Reference excerpt

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 {\displaystyle G} be the set Z 2 × Z 5 × Z 5 {\displaystyle \mathbb {Z} _{2}\times \mathbb {Z} _{5}\times \mathbb {Z} _{5}} . Define a binary operation ∘ {\displaystyle \circ } on G {\displaystyle G} such that for each ( a , b , c ) {\displaystyle (a,b,c)} and ( x , y , z ) {\displaystyle (x,y,z)} in G {\displaystyle G} ,

( a , b , c ) ∘ ( x , y , z ) = ( a + x , b − b x + y , c + ( − 1 ) a b y + 2 a z ) {\displaystyle (a,b,c)\circ (x,y,z)=(a+x,b-bx+y,c+(-1)^{a}by+2^{a}z)} . Then the Hoffman-Singleton graph has vertices g ∈ G {\displaystyle g\in G} and that there exists an edge between g ∈ G {\displaystyle g\in G} and g ′ ∈ G {\displaystyle g'\in G} whenever g ′ = g ∘ s {\displaystyle g'=g\circ s} for some s ∈ { ( 0 , 0 , 1 ) , ( 0 , 0 , 4 ) , ( 1 , 0 , 0 ) , ( 1 , 1 , 0 ) , ( 1 , 2 , 0 ) , ( 1 , 3 , 0 ) , ( 1 , 4 , 0 ) } {\displaystyle 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 b y {\displaystyle (-1)^{x}by} in the last term, but that does not produce the Hoffman-Singleton graph. It should instead be ( − 1 ) a b y {\displaystyle (-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

… excerpt ends here. Continue reading the full article.

Illustrations

Worked examples

Example 1 — a first encounter with Hoffman–Singleton graph

Start with the simplest possible case. Write down what Hoffman–Singleton graph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Hoffman–Singleton graph before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Hoffman–Singleton graph ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Hoffman–Singleton graph

In research: Hoffman–Singleton graph appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Hoffman–Singleton graph in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Hoffman–Singleton graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Individual graphs, Regular graphs, Strongly regular graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Hoffman–Singleton graph outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Hoffman–Singleton graph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hoffman–Singleton graph means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hoffman–Singleton graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hoffman–Singleton graph in simple terms?

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).

Why does Hoffman–Singleton graph matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Hoffman–Singleton graph?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Hoffman–Singleton graph.