Question

Give an example of each of the following: (i) an infinite graph with infinitely many end-vertices; (ii) an infinite graph with uncountably many vertices and edges.

Short answer

An infinite graph with infinitely many end-vertices can be represented by linking every pair of consecutive integers, resulting in a graph with vertices \(-\infty, -3, -2, -1, 0, 1, 2, 3, ...\), where zero is the only non-end-vertex. An infinite graph with uncountably many vertices and edges can be represented by the Real number line, where each real number is a vertex and there is an edge between every two real numbers.

Step-by-step solution

Understand graph terms

An end-vertex, also known as a leaf, is a vertex that has degree 1. An infinite graph is a graph with infinitely many vertices. A countable set is one where the elements can be paired one-to-one with the set of natural numbers. An uncountable set is one where such a pairing is not possible.

Infinite graph with infinitely many end-vertices

Consider the graph where the vertices are represented by integers, both positive and negative. Now join every pair of consecutive integers by an edge. This will result in a graph with vertices \(-\infty, -3, -2, -1, 0, 1, 2, 3, ...\), such that every vertex, except 0, is an end-vertex. Here, vertex 0 is the only vertex that is not an end-vertex since it has a degree of 2. This is because it is connected to -1 and 1.

Infinite graph with uncountably many vertices and edges

Let's consider the Real line to be the infinite graph with uncountably many vertices and edges. The vertices in this graph are all the real numbers, and there's an edge between every two real numbers. Real numbers form an uncountably infinite set. Thus, this graph has uncountably many vertices and edges because there's an edge between every two real numbers.

Key concepts

end-vertices

In graph theory, an end-vertex, also known as a leaf, is a special type of vertex. It is characterized by having exactly one edge connecting to it. Thus, it has a degree of 1.
To visualize an end-vertex, consider a tree branch. The leaf at the end of the branch only connects to one part of the tree.
In finite graphs, each leaf plays a crucial role in understanding the graph's structure. In infinite graphs, like the example provided with the integer line, end-vertices can extend infinitely.

• Each end-vertex in the integer line is connected to just one other integer.
• This makes all negative and positive integers, except for zero, end-vertices.
• Understanding end-vertices is important when analyzing infinite graphs.

countable set

A countable set is a concept arising in mathematics that helps to equate the size of infinite sets. A set is deemed countable if its elements can be matched one-to-one with the set of natural numbers.
Imagine lining up each element of the set and assigning a natural number to each. If every element can be paired uniquely with a natural number, the set is considered countable. This concept is crucial when dealing with infinite structures, as it allows us to understand their size and structure.

• For example, the set of all integers is countable.
• There is a method to pair each integer with a natural number, even if the set is infinite.
• Countable sets are foundational in understanding infinite graphs with infinite or infinite structures.

uncountable set

In contrast to countable sets, uncountable sets are sets that cannot be paired with the natural numbers. No matter how you try, you cannot form a one-to-one pairing. The most famous example of an uncountable set is the set of real numbers.
The concept of uncountability challenges our understanding of size in mathematics. It shows that some infinities are larger than others. This uncountability property can have significant implications, especially in graph theory, when creating graphs with an uncountable number of vertices and edges.

• The set of real numbers serves as a key example of an uncountable set.
• An uncountable set cannot be listed or sequentially arranged like a countable set.
• Graphs with uncountably many vertices are distinct due to their complex infinite nature.

real numbers

Real numbers include all the numbers on the number line—both rational and irrational numbers. They form an uncountable set. The properties of real numbers include continuity and density, which means between any two real numbers, there is always another real number.
This property of the real numbers forms the basis for creating uncountable graphs. If each real number represents a vertex and an edge connects every two real numbers, you derive a graph with uncountably many vertices and edges. Such a graph illustrates this concept vividly.

• Real numbers are used to represent continuous and dense data.
• Their uncountable nature makes them crucial in examples of uncountable graphs.
• Real number line graphs depict another level of infinity in graph theory.

degree of vertex

The degree of a vertex in graph theory describes the number of edges connected to it. Knowing the degree of each vertex helps in understanding the topology or arrangement of the graph.
The degree helps us discern special types of vertices, like end-vertices. End-vertices by definition have a degree of exactly one. In infinite graphs, analyzing degrees can be quite insightful, as it assists in identifying unique properties and relationships within a graph.

• The degree can vary from zero (isolated vertex) to any finite number, or even infinity.
• In the integer example, most vertices have degree one, except zero which demonstrates the role degree plays in distinguishing vertices.
• Recognizing vertex degree is crucial for deeper analysis in graph theory.