mathematics

Vertex-transitive graph

Vertex-transitive graph is a mathematics 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 Vertex-transitive graph rather than just read about it. In short: In the mathematical field of graph theory, an automorphism is a permutation of the vertices such that edges are mapped to edges and non-edges are mapped to non-edges. A graph is a vertex-transitive graph if, given any two vertices v1 and v2 of G, there is an automorphism f such that f ( v 1 ) = v 2 . {\displaystyle f(v_{1})=v_{2}.\ } In other words, a graph is vertex-transitive if its automorphism group acts transit…

Key takeaways

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

Reference excerpt

In the mathematical field of graph theory, an automorphism is a permutation of the vertices such that edges are mapped to edges and non-edges are mapped to non-edges. A graph is a vertex-transitive graph if, given any two vertices v1 and v2 of G, there is an automorphism f such that

f ( v 1 ) = v 2 . {\displaystyle f(v_{1})=v_{2}.\ }

In other words, a graph is vertex-transitive if its automorphism group acts transitively on its vertices. A graph is vertex-transitive if and only if its graph complement is, since the group actions are identical. Every symmetric graph without isolated vertices is vertex-transitive, and every vertex-transitive graph is regular. However, not all vertex-transitive graphs are symmetric (for example, the edges of the truncated tetrahedron), and not all regular graphs are vertex-transitive (for example, the Frucht graph and Tietze's graph).

Finite examples

Finite vertex-transitive graphs include the symmetric graphs (such as the Petersen graph, the Heawood graph and the vertices and edges of the Platonic solids). The finite Cayley graphs (such as cube-connected cycles) are also vertex-transitive, as are the vertices and edges of the Archimedean solids (though only two of these are symmetric). Potočnik, Spiga and Verret have constructed a census of all connected cubic vertex-transitive graphs on at most 1280 vertices. Although every Cayley graph is vertex-transitive, there exist other vertex-transitive graphs that are not Cayley graphs. The most famous example is the Petersen graph, but others can be constructed including the line graphs of edge-transitive non-bipartite graphs with odd vertex degrees.

Properties The edge-connectivity of a connected vertex-transitive graph is equal to the degree d, while the vertex-connectivity will be at least 2(d + 1)/3. If the degree is 4 or less, or the graph is also edge-transitive, or the graph is a minimal Cayley graph, then the vertex-connectivity will also be equal to d.

Infinite examples Infinite vertex-transitive graphs include:

infinite paths (infinite in both directions) infinite regular trees, e.g. the Cayley graph of the free group graphs of uniform tessellations (see a complete list of planar tessellations), including all tilings by regular polygons infinite Cayley graphs the Rado graph Two countable vertex-transitive graphs are called quasi-isometric if the ratio of their distance functions is bounded from below and from above. A well known conjecture stated that every infinite vertex-transitive graph is quasi-isometric to a Cayley graph. A counterexample was proposed by Diestel and Leader in 2001. In 2005, Eskin, Fisher, and Whyte confirmed the counterexample.

See also Edge-transitive graph Lovász conjecture Semi-symmetric graph Zero-symmetric graph

References

External links Weisstein, Eric W. "Vertex-transitive graph". MathWorld. A census of small connected cubic vertex-transitive graphs. Primož Potočnik, Pablo Spiga, Gabriel Verret, 2012. Vertex-transitive Graphs On Fewer Than 48 Vertices. Gordon Royle and Derek Holt, 2020.

Worked examples

Example 1 — a first encounter with Vertex-transitive graph

Start with the simplest possible case. Write down what Vertex-transitive graph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, 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 Vertex-transitive 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 Vertex-transitive 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 Vertex-transitive graph

In research: Vertex-transitive graph appears in mathematics 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 Vertex-transitive 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: Vertex-transitive graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic graph theory, Graph families, Regular graphs, so understanding it makes those chapters shorter.
In everyday life: Look for Vertex-transitive 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.

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How to study Vertex-transitive graph in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Vertex-transitive 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 Vertex-transitive graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Vertex-transitive graph in simple terms?

Why does Vertex-transitive graph matter?

Because it connects several mathematics 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 Vertex-transitive 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 Vertex-transitive graph.