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Loop (graph theory)

Loop (graph theory) 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 Loop (graph theory) rather than just read about it. In short: In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops.

Loop (graph theory) — main illustration
Loop (graph theory) — illustration

Key takeaways

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

Reference excerpt

In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops. Depending on the context, a graph or a multigraph may be defined so as to either allow or disallow the presence of loops (often in concert with allowing or disallowing multiple edges between the same vertices):

Where graphs are defined so as to allow loops and multiple edges, a graph without loops or multiple edges is often distinguished from other graphs by calling it a simple graph. Where graphs are defined so as to disallow loops and multiple edges, a graph that does have loops or multiple edges is often distinguished from the graphs that satisfy these constraints by calling it a multigraph or pseudograph. In a graph with one vertex, all edges must be loops. Such a graph is called a bouquet.

Degree For an undirected graph, the degree of a vertex is equal to the number of adjacent vertices. A special case is a loop, which adds two to the degree. This can be understood by letting each connection of the loop edge count as its own adjacent vertex. In other words, a vertex with a loop "sees" itself as an adjacent vertex from both ends of the edge thus adding two, not one, to the degree. For a directed graph, a loop adds one to the in degree and one to the out degree.

See also

In graph theory Cycle (graph theory) Graph theory Glossary of graph theory

In topology Möbius ladder Möbius strip Strange loop Klein bottle

References

Balakrishnan, V. K.; Graph Theory, McGraw-Hill; 1 edition (February 1, 1997). ISBN 0-07-005489-4. Bollobás, Béla; Modern Graph Theory, Springer; 1st edition (August 12, 2002). ISBN 0-387-98488-7. Diestel, Reinhard; Graph Theory, Springer; 2nd edition (February 18, 2000). ISBN 0-387-98976-5. Gross, Jonathon L, and Yellen, Jay; Graph Theory and Its Applications, CRC Press (December 30, 1998). ISBN 0-8493-3982-0. Gross, Jonathon L, and Yellen, Jay; (eds); Handbook of Graph Theory. CRC (December 29, 2003). ISBN 1-58488-090-2. Zwillinger, Daniel; CRC Standard Mathematical Tables and Formulae, Chapman & Hall/CRC; 31st edition (November 27, 2002). ISBN 1-58488-291-3.

External links This article incorporates public domain material from Paul E. Black. "Self loop". Dictionary of Algorithms and Data Structures. NIST.

Illustrations

Loop (graph theory): A graph with a loop on vertex 1
A graph with a loop on vertex 1

Worked examples

Example 1 — a first encounter with Loop (graph theory)

Start with the simplest possible case. Write down what Loop (graph theory) 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 Loop (graph theory) 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 Loop (graph theory) 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 Loop (graph theory)

In research
Loop (graph theory) 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 Loop (graph theory) 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
Loop (graph theory) is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph theory objects, so understanding it makes those chapters shorter.
In everyday life
Look for Loop (graph theory) 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 Loop (graph theory) in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Loop (graph theory) 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 Loop (graph theory) out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Loop (graph theory) in simple terms?

In graph theory, a loop (also called a self-loop or a buckle) is an edge that connects a vertex to itself. A simple graph contains no loops.

Why does Loop (graph theory) 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 Loop (graph theory)?

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 Loop (graph theory).

Tags

  • Graph theory objects

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