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Simplicial vertex

Simplicial vertex 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 Simplicial vertex rather than just read about it. In short: In graph theory, a simplicial vertex v {\displaystyle v} is a vertex whose closed neighborhood N G [ v ] {\displaystyle N_{G}[v]} in a graph G {\displaystyle G} forms a clique, where every pair of neighbors is adjacent to each other. A vertex of a graph is bisimplicial if the set of it and its neighbours is the union of two cliques, and is k-simplicial if the set is the union of k cliques.

Simplicial vertex — main illustration
Simplicial vertex — illustration

Key takeaways

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

Reference excerpt

In graph theory, a simplicial vertex v {\displaystyle v} is a vertex whose closed neighborhood N G [ v ] {\displaystyle N_{G}[v]} in a graph G {\displaystyle G} forms a clique, where every pair of neighbors is adjacent to each other. A vertex of a graph is bisimplicial if the set of it and its neighbours is the union of two cliques, and is k-simplicial if the set is the union of k cliques. A vertex is co-simplicial if its non-neighbours form an independent set. Addario-Berry et al. demonstrated that every even-hole-free graph (or more specifically, even-cycle-free graph, as 4-cycles are also excluded here) contains a bisimplicial vertex, which settled a conjecture by Reed. The proof was later shown to be flawed by Chudnovsky & Seymour, who gave a correct proof. Due to this property, the family of all even-cycle-free graphs is χ {\displaystyle \chi } -bounded.

See also Even-hole-free graph

χ {\displaystyle \chi } -bounded family of graphs

References

Illustrations

Simplicial vertex: Vertex 3 (circled red) is bisimplicial, as the set of it and its neighbors is the union of 2 cliques (denoted in black).
Vertex 3 (circled red) is bisimplicial, as the set of it and its neighbors is the union of 2 cliques (denoted in black).

Worked examples

Example 1 — a first encounter with Simplicial vertex

Start with the simplest possible case. Write down what Simplicial vertex 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 Simplicial vertex 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 Simplicial vertex 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 Simplicial vertex

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

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

Frequently asked questions

What is Simplicial vertex in simple terms?

In graph theory, a simplicial vertex v {\displaystyle v} is a vertex whose closed neighborhood N G [ v ] {\displaystyle N_{G}[v]} in a graph G {\displaystyle G} forms a clique, where every pair of neighbors is adjacent to each other. A vertex of a graph is bisimplicial if the set of it and its neig…

Why does Simplicial vertex 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 Simplicial vertex?

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 Simplicial vertex.

Tags

  • Graph theory
  • Graph theory stubs

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