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Graph enumeration

Graph enumeration is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Graph enumeration rather than just read about it. In short: In combinatorics, an area of mathematics, graph enumeration describes a class of combinatorial enumeration problems in which one must count undirected or directed graphs of certain types, typically as a function of the number of vertices of the graph. These problems may be solved either exactly (as an algebraic enumeration problem) or asymptotically.

Graph enumeration — main illustration
Graph enumeration — illustration

Key takeaways

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

Reference excerpt

In combinatorics, an area of mathematics, graph enumeration describes a class of combinatorial enumeration problems in which one must count undirected or directed graphs of certain types, typically as a function of the number of vertices of the graph. These problems may be solved either exactly (as an algebraic enumeration problem) or asymptotically. The pioneers in this area of mathematics were George Pólya, Arthur Cayley and J. Howard Redfield.

Labeled vs unlabeled problems In some graphical enumeration problems, the vertices of the graph are considered to be labeled in such a way as to be distinguishable from each other, while in other problems any permutation of the vertices is considered to form the same graph, so the vertices are considered identical or unlabeled. In general, labeled problems tend to be easier. As with combinatorial enumeration more generally, the Pólya enumeration theorem is an important tool for reducing unlabeled problems to labeled ones: each unlabeled class is considered as a symmetry class of labeled objects. The number of unlabelled graphs with n {\displaystyle n} vertices is still not known in a closed-form solution, but as almost all graphs are asymmetric this number is asymptotic to

2 ( n 2 ) n ! . {\displaystyle {\frac {2^{\tbinom {n}{2}}}{n!}}.}

Exact enumeration formulas Some important results in this area include the following.

The number of labeled n-vertex simple undirected graphs is 2n(n−1)/2. The number of labeled n-vertex simple directed graphs is 2n(n−1). The number Cn of connected labeled n-vertex undirected graphs satisfies the recurrence relation

C n = 2 ( n 2 ) − 1 n ∑ k = 1 n − 1 k ( n k ) 2 ( n − k 2 ) C k . {\displaystyle C_{n}=2^{n \choose 2}-{\frac {1}{n}}\sum _{k=1}^{n-1}k{n \choose k}2^{n-k \choose 2}C_{k}.}

from which one may easily calculate, for n = 1, 2, 3, ..., that the values for Cn are 1, 1, 4, 38, 728, 26704, 1866256, ...(sequence A001187 in the OEIS) The number of labeled n-vertex free trees is nn−2 (Cayley's formula). The number of unlabeled n-vertex caterpillars is

2 n − 4 + 2 ⌊ ( n − 4 ) / 2 ⌋ . {\displaystyle 2^{n-4}+2^{\lfloor (n-4)/2\rfloor }.}

Graph database Various research groups have provided searchable database that lists graphs with certain properties of a small sizes. For example

The House of Graphs Small Graph Database

References

Illustrations

Graph enumeration: The complete list of all free trees on 2, 3, and 4 labeled vertices: 2 2 − 2 = 1 {\displaystyle 2^{2-2}=1} tree with 2 vertices, 3 3 − 2 = 3 {\displaystyle 3^{3-2}=3} trees with 3 vertices, and 4 4 − 2 = 16 {\displaystyle 4^{4-2}=16} trees with 4 vertices.
The complete list of all free trees on 2, 3, and 4 labeled vertices: 2 2 − 2 = 1 {\displaystyle 2^{2-2}=1} tree with 2 vertices, 3 3 − 2 = 3 {\displaystyle 3^{3-2}=3} trees with 3 vertices, and 4 4 − 2 = 16 {\displaystyle 4^{4-2}=16} trees with 4 vertices.

Worked examples

Example 1 — a first encounter with Graph enumeration

Start with the simplest possible case. Write down what Graph enumeration 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 Graph enumeration 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 Graph enumeration 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 Graph enumeration

In research
Graph enumeration 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 Graph enumeration 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
Graph enumeration is common in secondary-school and first-year university syllabi. It links to neighbouring topics Enumerative combinatorics, Graph enumeration, so understanding it makes those chapters shorter.
In everyday life
Look for Graph enumeration 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.
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How to study Graph enumeration in 20 minutes

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

Frequently asked questions

What is Graph enumeration in simple terms?

In combinatorics, an area of mathematics, graph enumeration describes a class of combinatorial enumeration problems in which one must count undirected or directed graphs of certain types, typically as a function of the number of vertices of the graph. These problems may be solved either exactly (as…

Why does Graph enumeration 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 Graph enumeration?

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 Graph enumeration.

Tags

  • Enumerative combinatorics
  • Graph enumeration

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