science

Ljubljana graph

Ljubljana graph 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 Ljubljana graph rather than just read about it. In short: In the mathematical field of graph theory, the Ljubljana graph is an undirected bipartite graph with 112 vertices and 168 edges, rediscovered in 2002 and named after Ljubljana (the capital of Slovenia). It is a cubic graph with diameter 8, radius 7, chromatic number 2 and chromatic index 3.

Key takeaways

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

Reference excerpt

In the mathematical field of graph theory, the Ljubljana graph is an undirected bipartite graph with 112 vertices and 168 edges, rediscovered in 2002 and named after Ljubljana (the capital of Slovenia). It is a cubic graph with diameter 8, radius 7, chromatic number 2 and chromatic index 3. Its girth is 10 and there are exactly 168 cycles of length 10 in it. There are also 168 cycles of length 12.

Construction The Ljubljana graph is Hamiltonian and can be constructed from the LCF notation : [47, −23, −31, 39, 25, −21, −31, −41, 25, 15, 29, −41, −19, 15, −49, 33, 39, −35, −21, 17, −33, 49, 41, 31, −15, −29, 41, 31, −15, −25, 21, 31, −51, −25, 23, 9, −17, 51, 35, −29, 21, −51, −39, 33, −9, −51, 51, −47, −33, 19, 51, −21, 29, 21, −31, −39]2. The Ljubljana graph is the Levi graph of the Ljubljana configuration, a quadrangle-free configuration with 56 lines and 56 points. In this configuration, each line contains exactly 3 points, each point belongs to exactly 3 lines and any two lines intersect in at most one point.

Algebraic properties The automorphism group of the Ljubljana graph is a group of order 168. It acts transitively on the edges the graph but not on its vertices: there are symmetries taking every edge to any other edge, but not taking every vertex to any other vertex. Therefore, the Ljubljana graph is a semi-symmetric graph, the third smallest possible cubic semi-symmetric graph after the Gray graph on 54 vertices and the Iofinova-Ivanov graph on 110 vertices. The characteristic polynomial of the Ljubljana graph is

( x − 3 ) x 14 ( x + 3 ) ( x 2 − x − 4 ) 7 ( x 2 − 2 ) 6 ( x 2 + x − 4 ) 7 ( x 4 − 6 x 2 + 4 ) 14 . {\displaystyle (x-3)x^{14}(x+3)(x^{2}-x-4)^{7}(x^{2}-2)^{6}(x^{2}+x-4)^{7}(x^{4}-6x^{2}+4)^{14}.\ }

History The Ljubljana graph was first published in 1993 by Brouwer, Dejter and Thomassen as a self-complementary subgraph of the Dejter graph. In 1972, Bouwer was already talking of a 112-vertices edge- but not vertex-transitive cubic graph found by R. M. Foster, nonetheless unpublished. Conder, Malnič, Marušič, Pisanski and Potočnik rediscovered this 112-vertices graph in 2002 and named it the Ljubljana graph after the capital of Slovenia. They proved that it was the unique 112-vertices edge- but not vertex-transitive cubic graph and therefore that was the graph found by Foster.

Gallery

References

Illustrations

Worked examples

Example 1 — a first encounter with Ljubljana graph

Start with the simplest possible case. Write down what Ljubljana graph 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 Ljubljana 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 Ljubljana 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 Ljubljana graph

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

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

Frequently asked questions

What is Ljubljana graph in simple terms?

Why does Ljubljana graph 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 Ljubljana 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 Ljubljana graph.