science

Graph operations

Graph operations 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 operations rather than just read about it. In short: In the mathematical field of graph theory, graph operations are operations which produce new graphs from initial ones. They include both unary (one input) and binary (two input) operations.

Key takeaways

Graph operations 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 operations to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Graph operations from memory before moving on to harder problems.

Reference excerpt

In the mathematical field of graph theory, graph operations are operations which produce new graphs from initial ones. They include both unary (one input) and binary (two input) operations.

Unary operations Unary operations create a new graph from a single initial graph.

Elementary operations Elementary operations or editing operations, which are also known as graph edit operations, create a new graph from one initial one by a simple local change, such as addition or deletion of a vertex or of an edge, merging and splitting of vertices, edge contraction, etc. The graph edit distance between a pair of graphs is the minimum number of elementary operations required to transform one graph into the other.

Advanced operations Advanced operations create a new graph from an initial one by a complex change, such as:

transpose graph; complement graph; line graph; graph minor; graph rewriting; power of graph; dual graph; medial graph; quotient graph; double graph; simplex graph; YΔ- and ΔY-transformation; Mycielskian.

Binary operations Binary operations create a new graph from two initial graphs G1 = (V1, E1) and G2 = (V2, E2), such as:

graph union: G1 ∪ G2. There are two definitions. In the most common one, the disjoint union of graphs, the union is assumed to be disjoint. Less commonly (though more consistent with the general definition of union in mathematics) the union of two graphs is defined as the graph (V1 ∪ V2, E1 ∪ E2). graph intersection: G1 ∩ G2 = (V1 ∩ V2, E1 ∩ E2); graph join: G 1 ∇ G 2 {\displaystyle G_{1}\nabla G_{2}} . Graph with all the edges that connect the vertices of the first graph with the vertices of the second graph. It is a commutative operation (for unlabelled graphs); graph products based on the cartesian product of the vertex sets: cartesian graph product: it is a commutative and associative operation (for unlabelled graphs), lexicographic graph product (or graph composition): it is an associative (for unlabelled graphs) and non-commutative operation, strong graph product: it is a commutative and associative operation (for unlabelled graphs), tensor graph product (or direct graph product, categorical graph product, cardinal graph product, Kronecker graph product): it is a commutative and associative operation (for unlabelled graphs), replacement product, zig-zag graph product; graph product based on other products: rooted graph product: it is an associative operation (for unlabelled but rooted graphs), corona graph product: it is a non-commutative operation; series–parallel graph composition: parallel graph composition: it is a commutative operation (for unlabelled graphs), series graph composition: it is a non-commutative operation, source graph composition: it is a commutative operation (for unlabelled graphs); Hajós construction.

Notes

Worked examples

Example 1 — a first encounter with Graph operations

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

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

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

Frequently asked questions

What is Graph operations in simple terms?

In the mathematical field of graph theory, graph operations are operations which produce new graphs from initial ones. They include both unary (one input) and binary (two input) operations.

Why does Graph operations 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 operations?

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 operations.