science

Magic graph

Magic 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 Magic graph rather than just read about it. In short: A magic graph is a graph whose edges are labelled by the first q positive integers, where q is the number of edges, so that the sum over the edges incident with any vertex is the same, independent of the choice of vertex; or it is a graph that has such a labelling. The name "magic" sometimes means that the integers are any positive integers; then the graph and the labelling using the first q positive integers are ca…

Key takeaways

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

Reference excerpt

A magic graph is a graph whose edges are labelled by the first q positive integers, where q is the number of edges, so that the sum over the edges incident with any vertex is the same, independent of the choice of vertex; or it is a graph that has such a labelling. The name "magic" sometimes means that the integers are any positive integers; then the graph and the labelling using the first q positive integers are called supermagic. A graph is vertex-magic if its vertices can be labelled so that the sum on any edge is the same. It is total magic if its edges and vertices can be labelled so that the vertex label plus the sum of labels on edges incident with that vertex is a constant. There are a great many variations on the concept of magic labelling of a graph. There is much variation in terminology as well. The definitions here are perhaps the most common. Comprehensive references for magic labellings and magic graphs are Gallian (1998), Wallis (2001), and Marr and Wallis (2013).

Magic squares

A semimagic square is an n × n square with the numbers 1 to n2 in its cells, in which the sum of each row and column is the same. A semimagic square is equivalent to a magic labelling of the complete bipartite graph Kn,n. The two vertex sets of Kn,n correspond to the rows and the columns of the square, respectively, and the label on an edge risj is the value in row i, column j of the semimagic square. The definition of semimagic squares differs from the definition of magic squares in the treatment of the diagonals of the square. Magic squares are required to have diagonals with the same sum as the row and column sums, but for semimagic squares this is not required. Thus, every magic square is semimagic, but not vice versa.

References Nora Hartsfield and Gerhard Ringel (1994, 2003), Pearls in Graph Theory, revised edition. Dover Publications, Mineola, N.Y. Section 6.1. W. D. Wallis (2001), Magic Graphs. Birkhäuser Boston, Boston, Mass. ISBN 0-8176-4252-8 Alison M. Marr and W. D. Wallis (2013), Magic Graphs. Second edition. Birkhäuser/Springer, New York. ISBN 978-0-8176-8390-0; 978-0-8176-8391-7 Joseph A. Gallian (1998), A dynamic survey of graph labeling. Electronic Journal of Combinatorics, vol. 5, Dynamic Survey 6. Updated many times.

Worked examples

Example 1 — a first encounter with Magic graph

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

In research: Magic 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 Magic 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: Magic graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Graph theory stubs, Graphs, Magic figures, so understanding it makes those chapters shorter.
In everyday life: Look for Magic 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.

Affiliate

Preply — study more efficiently by working with a personal tutor. 50% off.

How to study Magic graph in 20 minutes

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

Frequently asked questions

What is Magic graph in simple terms?

Why does Magic 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 Magic 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 Magic graph.