science

Null graph

Null 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 Null graph rather than just read about it. In short: In the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph"). Order-zero graph The order-zero graph, K0, is the unique graph having no vertices (hence its order is zero).

Key takeaways

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

Reference excerpt

In the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph").

Order-zero graph

The order-zero graph, K0, is the unique graph having no vertices (hence its order is zero). It follows that K0 also has no edges. Thus the null graph is a regular graph of degree zero. Some authors exclude K0 from consideration as a graph (either by definition, or more simply as a matter of convenience). Whether including K0 as a valid graph is useful depends on context. On the positive side, K0 follows naturally from the usual set-theoretic definitions of a graph (it is the ordered pair (V, E) for which the vertex and edge sets, V and E, are both empty), in proofs it serves as a natural base case for mathematical induction, and similarly, in recursively defined data structures K0 is useful for defining the base case for recursion (by treating the null tree as the child of missing edges in any non-null binary tree, every non-null binary tree has exactly two children). On the negative side, including K0 as a graph requires that many well-defined formulas for graph properties include exceptions for it (for example, either "counting all strongly connected components of a graph" becomes "counting all non-null strongly connected components of a graph", or the definition of connected graphs has to be modified not to include K0). To avoid the need for such exceptions, it is often assumed in literature that the term graph implies "graph with at least one vertex" unless context suggests otherwise. In category theory, the order-zero graph is, according to some definitions of "category of graphs," the initial object in the category. K0 does fulfill (vacuously) most of the same basic graph properties as does K1 (the graph with one vertex and no edges). As some examples, K0 is of size zero, it is equal to its complement graph K0, a forest, and a planar graph. It may be considered undirected, directed, or even both; when considered as directed, it is a directed acyclic graph. And it is both a complete graph and an edgeless graph. However, definitions for each of these graph properties will vary depending on whether context allows for K0.

Edgeless graph

For each natural number n, the edgeless graph (or empty graph) Kn of order n is the graph with n vertices and zero edges. An edgeless graph is occasionally referred to as a null graph in contexts where the order-zero graph is not permitted. It is a 0-regular graph. The notation Kn arises from the fact that the n-vertex edgeless graph is the complement of the complete graph Kn.

See also Glossary of graph theory Cycle graph Path graph

Notes

References

External links Media related to Null graphs at Wikimedia Commons

Illustrations

Worked examples

Example 1 — a first encounter with Null graph

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

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

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

Frequently asked questions

What is Null graph in simple terms?

In the mathematical field of graph theory, the term "null graph" may refer either to the order-zero graph, or alternatively, to any edgeless graph (the latter is sometimes called an "empty graph"). Order-zero graph The order-zero graph, K0, is the unique graph having no vertices (hence its order is…

Why does Null 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 Null 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 Null graph.