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Highly irregular graph

Highly irregular 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 Highly irregular graph rather than just read about it. In short: In graph theory, a highly irregular graph is a graph in which, for every vertex, all neighbors of that vertex have distinct degrees. History Irregular graphs were initially characterized by Yousef Alavi, Gary Chartrand, Fan Chung, Paul Erdős, Ronald Graham, and Ortrud Oellermann.

Highly irregular graph — main illustration
Highly irregular graph — illustration

Key takeaways

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

Reference excerpt

In graph theory, a highly irregular graph is a graph in which, for every vertex, all neighbors of that vertex have distinct degrees.

History Irregular graphs were initially characterized by Yousef Alavi, Gary Chartrand, Fan Chung, Paul Erdős, Ronald Graham, and Ortrud Oellermann. They were motivated to define the 'opposite' of a regular graph, a concept which has been thoroughly studied and well understood.

Locality and regularity Defining an 'irregular graph' was not immediately obvious. In a k-regular graph, all vertices have degree k. In any graph G with more than one vertex, two vertices in G must have the same degree, so an irregular graph cannot be defined as a graph with all vertices of different degrees. One may be tempted then to define an irregular graph as having all vertices of distinct degrees except for two, but these types of graphs are also well understood and thus not interesting. Graph theorists thus turned to the issue of local regularity. A graph is locally regular at a vertex v if all vertices adjacent to v have degree r. A graph is thus locally irregular if for each vertex v of G the neighbors of v have distinct degrees, and these graphs are thus termed highly irregular graphs.

Properties of irregular graphs Some facts about highly irregular graphs outlined by Alavi et al.:

If v is a vertex of maximum degree d in a highly irregular graph H, then v is adjacent to exactly one vertex of each degree 1, 2, ..., d. The largest degree in a highly irregular graph is at most half the number of vertices. If H is a highly irregular graph with maximum degree d, one can construct a highly irregular graph of degree d+1 by taking two copies of H and adding an edge between the two vertices of degree d. H(n)/G(n) goes to 0 as n goes to infinity exponentially rapidly, where H(n) is the number of (non-isomorphic) highly irregular graphs with n vertices, and G(n) is the total number of graphs with n vertices. For every graph G, there exists a highly irregular graph H containing G as an induced subgraph. This last observation is analogous to a result of Dénes Kőnig, which states that if H is a graph with greatest degree r, then there is a graph G which is r-regular and contains H as an induced subgraph.

Applications of irregularity Definitions of irregularity have been important in the study of network heterogeneity, which has implications in networks found across biology, ecology, technology, and economy. There have been several graph statistics that have been suggested, many of which are based on the number of vertices in a graph and their degrees. The characterization of highly irregular graphs has also been applied to the question of heterogeneity, yet all of these fail to shed enough light on real-world situations. Efforts continue to be made to find appropriate ways to quantify network heterogeneity.

See also Stepwise irregular graph Imbalance conjecture

References

Illustrations

Highly irregular graph: Each vertex has neighbors with unique degrees. For example, the black vertex has the yellow vertices as its neighbors, which have degrees 1, 2 and 3.
Each vertex has neighbors with unique degrees. For example, the black vertex has the yellow vertices as its neighbors, which have degrees 1, 2 and 3.

Worked examples

Example 1 — a first encounter with Highly irregular graph

Start with the simplest possible case. Write down what Highly irregular 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 Highly irregular 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 Highly irregular 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 Highly irregular graph

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

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

Frequently asked questions

What is Highly irregular graph in simple terms?

In graph theory, a highly irregular graph is a graph in which, for every vertex, all neighbors of that vertex have distinct degrees. History Irregular graphs were initially characterized by Yousef Alavi, Gary Chartrand, Fan Chung, Paul Erdős, Ronald Graham, and Ortrud Oellermann.

Why does Highly irregular 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 Highly irregular 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 Highly irregular graph.

Tags

  • Graph families

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