science

Lexicographic product of graphs

Lexicographic product of graphs 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 Lexicographic product of graphs rather than just read about it. In short: In graph theory, the lexicographic product or (graph) composition G ∙ H of graphs G and H is a graph such that the vertex set of G ∙ H is the cartesian product V(G) × V(H); and any two vertices (u,v) and (x,y) are adjacent in G ∙ H if and only if either u is adjacent to x in G or u = x and v is adjacent to y in H. If the edge relations of the two graphs are order relations, then the edge relation of their lexicograp…

Key takeaways

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

Reference excerpt

In graph theory, the lexicographic product or (graph) composition G ∙ H of graphs G and H is a graph such that

the vertex set of G ∙ H is the cartesian product V(G) × V(H); and any two vertices (u,v) and (x,y) are adjacent in G ∙ H if and only if either u is adjacent to x in G or u = x and v is adjacent to y in H. If the edge relations of the two graphs are order relations, then the edge relation of their lexicographic product is the corresponding lexicographic order. It is one of 4 common graph products including Cartesian, tensor, and strong. The lexicographic product was first studied by Felix Hausdorff (1914). As Feigenbaum & Schäffer (1986) showed, the problem of recognizing whether a graph is a lexicographic product is equivalent in complexity to the graph isomorphism problem.

Properties The lexicographic product is in general noncommutative: G ∙ H ≠ H ∙ G. However it satisfies a distributive law with respect to disjoint union: (A + B) ∙ C = A ∙ C + B ∙ C. In addition it satisfies an identity with respect to complementation: C(G ∙ H) = C(G) ∙ C(H). In particular, the lexicographic product of two self-complementary graphs is self-complementary. The independence number of a lexicographic product may be easily calculated from that of its factors:

α(G ∙ H) = α(G)α(H). The clique number of a lexicographic product is as well multiplicative:

ω(G ∙ H) = ω(G)ω(H). The chromatic number of a lexicographic product is equal to the b-fold chromatic number of G, for b equal to the chromatic number of H:

χ(G ∙ H) = χb(G), where b = χ(H). The lexicographic product of two graphs is a perfect graph if and only if both factors are perfect.

Notes

References Feigenbaum, J.; Schäffer, A. A. (1986), "Recognizing composite graphs is equivalent to testing graph isomorphism", SIAM Journal on Computing, 15 (2): 619–627, doi:10.1137/0215045, MR 0837609. Geller, D.; Stahl, S. (1975), "The chromatic number and other functions of the lexicographic product", Journal of Combinatorial Theory, Series B, 19: 87–95, doi:10.1016/0095-8956(75)90076-3, MR 0392645. Hausdorff, F. (1914), Grundzüge der Mengenlehre, Leipzig{{citation}}: CS1 maint: location missing publisher (link) Imrich, Wilfried; Klavžar, Sandi (2000), Product Graphs: Structure and Recognition, Wiley, ISBN 0-471-37039-8 Ravindra, G.; Parthasarathy, K. R. (1977), "Perfect product graphs", Discrete Mathematics, 20 (2): 177–186, doi:10.1016/0012-365X(77)90056-5, hdl:10338.dmlcz/102469, MR 0491304.

External links Weisstein, Eric W. "Graph Lexicographic Product". MathWorld.

Illustrations

Worked examples

Example 1 — a first encounter with Lexicographic product of graphs

Start with the simplest possible case. Write down what Lexicographic product of graphs 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 Lexicographic product of graphs 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 Lexicographic product of graphs 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 Lexicographic product of graphs

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

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Frequently asked questions

What is Lexicographic product of graphs in simple terms?

In graph theory, the lexicographic product or (graph) composition G ∙ H of graphs G and H is a graph such that the vertex set of G ∙ H is the cartesian product V(G) × V(H); and any two vertices (u,v) and (x,y) are adjacent in G ∙ H if and only if either u is adjacent to x in G or u = x and v is adj…

Why does Lexicographic product of graphs 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 Lexicographic product of graphs?

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 Lexicographic product of graphs.