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Chapter 14: Problem 57

Give an example of a tree with six vertices whose degrees are \(1,1,2,2,2\), and 2 .

Short Answer

Expert verified
One example of a tree with six vertices whose degrees are \(1,1,2,2,2,2\) is a simple path with four vertices (A-B-C-D) and two additional leaf vertices (E attached to A and F attached to D).

Step by step solution

01

Analyze the degrees

First, take a look at the given degrees which are \(1,1,2,2,2,2\). We have six degrees, indicating six vertices. The vertices with degree 1 are leaf nodes (end vertices) attached to only one edge. The vertices with degree 2 are inside nodes, each attached to two edges.
02

Draw the vertices with degree 2

Begin by drawing four vertices (named for instance as A, B, C and D) for the four vertices each with degree 2. Connect these vertices such that they form a path A-B-C-D. These four vertices form the backbone of our tree.
03

Assign the vertices with degree 1

Lastly, draw two more vertices (named for instance as E and F) representing the two vertices with degree 1. Connect one of them to vertex A and the other to vertex D. These vertices are the leaf nodes of our tree.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graph Theory
Graph theory is a field of mathematics that studies the relationships between pairs of objects. These objects are referred to as vertices, and the connections between them are called edges. Graphs can take on a variety of forms, including paths, trees, cycles, and more complex structures. A tree is a special type of graph that serves as a useful concept here.

  • A tree is a connected graph without cycles.
  • It has a hierarchical structure and is often used to represent organizational forms, computer networking, and data structures.
In the context of our example, the assignment involves creating a tree graph with specific sequences of degrees assigned to its vertices. Understanding graph theory helps to visualize and systematize relationships in structured formats.
Degree of Vertices
The degree of a vertex in a graph refers to the number of edges meeting at that vertex. In simpler words, the degree is how many connections a particular vertex has with other vertices. In our example, the degrees are given as 1, 1, 2, 2, 2, and 2.

  • Vertices with degree 1 are usually called leaf nodes.
  • Vertices with degree 2 in this example form the main path of the tree.
Understanding the degree of each vertex helps us shape the structure of the tree and decide how to interconnect the vertices. Ensuring that the sum of all degrees is twice the total number of edges gives a useful check when forming trees.
Leaf Nodes
Leaf nodes, also known as terminal nodes, are vertices whose degree is exactly 1. These nodes are the endpoints of the tree structure. Their role is crucial because they do not form a part of the main structure but rather terminate paths.

In our example:
  • We have two leaf nodes with degree 1.
  • These are separately attached to vertices in the tree like decorations at the ends, confirming their roles as endpoints.
Leaf nodes give a tree its final structure and often its visual appeal, highlighting both its span and end limits. They are a critical component in delineating the boundaries of paths within a tree.
Path
A path in graph theory refers to a sequence of vertices where each consecutive pair is connected by an edge. In the simplest terms, it's how you can "walk" from one vertex to another following the edges.

For the tree in our example:
  • We form a path with vertices that have degree of 2, creating the path A-B-C-D.
  • This path becomes the backbone of our tree, establishing the main structure.
Paths ensure that all vertices are accessible and connected, maintaining the integrity of the tree structure without forming loops. By controlling the path's formation, we can effectively manage how information or relationships are distributed through the charted structure. Understanding paths is fundamental to using graph theory to build logical and organized systems.

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Most popular questions from this chapter

Use a tree to model the employee relationships among the chief administrators of a large community college system: Three campus vice presidents report directly to the college president. On two campuses, the academic dean, the dean for administration, and the dean of student services report directly to the vice president. On the third campus, only the academic dean and the dean for administration report directly to the vice president.

Draw two equivalent graphs for each description. The vertices are \(A, B, C\), and \(D\). The edges are \(A D, B C, D C\), \(B B\), and \(D B\)

What is a tree?

Eight students form a math homework group. The students in the group are Zeb, Stryder, Amy, Jed, Evito, Moray, Carrie, and Oryan. Prior to forming the group, Stryder was friends with everyone but Moray. Moray was friends with Zeb, Amy, Carrie, and Evito. Jed was friends with Stryder, Evito, Oryan, and Zeb. Draw a graph that models pairs of friendships among the eight students prior to forming the math homework group.

A connected graph is described. Determine whether the graph has an Euler path (but not an Euler circuit), an Euler circuit, or neither an Euler path nor an Euler circuit. Explain your answer. The graph has 77 even vertices and four odd vertices.
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