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Integral graph

Integral 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 Integral graph rather than just read about it. In short: In the mathematical field of graph theory, an integral graph is a graph whose adjacency matrix's spectrum consists entirely of integers. In other words, a graph is an integral graph if all of the roots of the characteristic polynomial of its adjacency matrix are integers.

Integral graph — main illustration
Integral graph — illustration

Key takeaways

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

Reference excerpt

In the mathematical field of graph theory, an integral graph is a graph whose adjacency matrix's spectrum consists entirely of integers. In other words, a graph is an integral graph if all of the roots of the characteristic polynomial of its adjacency matrix are integers. The notion was introduced in 1974 by Frank Harary and Allen Schwenk.

Examples The complete graph Kn is integral for all n. The only cycle graphs that are integral are C 3 {\displaystyle C_{3}} , C 4 {\displaystyle C_{4}} , and C 6 {\displaystyle C_{6}} . If a graph is integral, then so is its complement graph; for instance, the complements of complete graphs, edgeless graphs, are integral. If two graphs are integral, then so is their Cartesian product and strong product; for instance, the Cartesian products of two complete graphs, the rook's graphs, are integral. Similarly, the hypercube graphs, as Cartesian products of any number of complete graphs K 2 {\displaystyle K_{2}} , are integral. The line graph of a regular integral graph is again integral. For instance, as the line graph of K 4 {\displaystyle K_{4}} , the octahedral graph is integral, and as the complement of the line graph of K 5 {\displaystyle K_{5}} , the Petersen graph is integral. Among the cubic symmetric graphs the utility graph, the Petersen graph, the Nauru graph and the Desargues graph are integral. The Higman–Sims graph, the Hall–Janko graph, the Clebsch graph, the Hoffman–Singleton graph, the Shrikhande graph and the Hoffman graph are integral. A regular graph is periodic if and only if it is an integral graph. A walk-regular graph that admits perfect state transfer is an integral graph. The Sudoku graphs, graphs whose vertices represent cells of a Sudoku board and whose edges represent cells that should not be equal, are integral.

References

Illustrations

Integral graph: The blue graph, C4, is one of the only integral cycle graphs, whose adjacency matrix has eigenvalues 0 , 0 , 2 , − 2 {\displaystyle 0,0,2,-2} . The red graph is not integral, as its eigenvalues are 0 , − 1 , 17 + 1 2 , − 17 + 1 2 {\displaystyle 0,-1,{\frac {{\sqrt {17}}+1}{2}},{\frac {-{\sqrt {17}}+1}{2}}} .
The blue graph, C4, is one of the only integral cycle graphs, whose adjacency matrix has eigenvalues 0 , 0 , 2 , − 2 {\displaystyle 0,0,2,-2} . The red graph is not integral, as its eigenvalues are 0 , − 1 , 17 + 1 2 , − 17 + 1 2 {\displaystyle 0,-1,{\frac {{\sqrt {17}}+1}{2}},{\frac {-{\sqrt {17}}+1}{2}}} .

Worked examples

Example 1 — a first encounter with Integral graph

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

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

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Integral graph means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Integral graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Integral graph in simple terms?

In the mathematical field of graph theory, an integral graph is a graph whose adjacency matrix's spectrum consists entirely of integers. In other words, a graph is an integral graph if all of the roots of the characteristic polynomial of its adjacency matrix are integers.

Why does Integral 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 Integral 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 Integral graph.

Tags

  • Algebraic graph theory
  • Graph families

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