/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 16 Draw all nonisomorphic rooted tr... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 9: Problem 16

Draw all nonisomorphic rooted trees having three vertices.

Short Answer

Expert verified
The exercise's solution is two trees. The first tree has a root with two branches extending separately from it, forming a 'V' shape. The second tree has a linear configuration with the root vertex connected to a second vertex, which is then connected to a third vertex.

Step by step solution

01

Understanding the Problem

A tree graph is a type of graph in which any two vertices are connected by exactly one path. This means there are no loops in tree graphs. A non-isomorphic graph has a unique graph commutation class. This means non-isomorphic graphs cannot be made to correlate with each other just by re-drawing. Thus, all nonisomorphic rooted trees having three vertices are those trees with three vertices (a root and two other vertices) that do not have the same shape. Understanding the above definitions could aid in solving the exercise.
02

Drawing the rooted trees

In order to create a rooted tree with three vertices, two vertices must be connected to the root. Remember that the tree cannot have any loops, and no two trees can be isomorphic. Draw the first tree with one vertex as the root and the remaining two vertices branching out individually from the root. Draw the second possible tree with one vertex as the root, another connected directly to the root, and the third connected to the second vertex. The first tree forms a 'V' shape with the root at the bottom, while the second forms a linear shape.
03

Final Confirmation

After drawing the two possible rooted trees with three vertices, inspect them to ensure they are non-isomorphic. The given trees are clearly different, not just a re-drawing of the same graph. Therefore, these are the two nonisomorphic, rooted trees with three vertices.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Nonisomorphic Graphs
Understanding the concept of nonisomorphic graphs is imperative when dealing with tree graphs. These are graphs that cannot be transformed into one another simply by relabeling or reorienting their vertices.
Nonisomorphic graphs might have the same number of vertices and edges, but their arrangements differ entirely:
  • The structures do not align when one graph is twisted or turned to match another.
  • No matter how the vertices are rearranged, the connectivity pattern must remain unaltered.
  • This makes nonisomorphic graphs unique in terms of their configuration.
Distinguishing between nonisomorphic graphs comes down to examining their structures closely to understand if they can be morphed into each other through any transformation.
Tree Graphs
Tree graphs are a special subset of graphs that are incredibly simple yet crucial for understanding hierarchical structures in mathematics and computer science.
Tree graphs have some foundational properties:
  • They are connected: All vertices in a tree graph are linked in such a way that there is only one path between any pair of vertices.
  • There are no loops: This means there are no circular paths in the graph.
  • They follow the equation: For a tree graph with \(n\) vertices, there will always be \(n-1\) edges.
Each tree graph is acyclic, meaning it doesn't contain any cycles, and each edge adds a new possibility to connect existing vertices straightforwardly, without forming loops.
Vertices in Graphs
Vertices are the fundamental units of any graph, including tree graphs, acting as nodes or points that stay connected through edges.
In tree graphs, vertices play vital roles:
  • A vertex can be the root or any other node in the graph structure.
  • The root vertex is distinct and serves as the starting point or top node in a rooted tree.
  • All paths in the tree graph stem from the root, ensuring every other vertex is connected directly or indirectly to it.
Understanding vertices is key to drawing and analyzing tree graphs effectively, as they are the building blocks that determine the shape and structure of the graph.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Write a recursive algorithm that initializes each vertex of a binary tree to the number of its descendants.

Can the first player always tie in Mu Torere?

Let \(G\) be a connected, weighted graph. Show that if, as long as possible, we remove an edge from \(G\) having maximum weight whose removal does not disconnect \(G,\) the result is a minimal spanning tree for \(G\).

Evaluate each vertex in each game tree. The values of the terminal vertices are given.

Draw the complete game tree for nim in which the initial position consists of two piles, one containing three tokens and the other containing two tokens. Assume that the last player to take a token wins. Assign values to all vertices so that the resulting tree is analogous to Figure \(9.9 .2 .\) Will the first or second player, playing an optimal strategy, always win? Describe an optimal strategy for the winning player.
See all solutions

Recommended explanations on Math Textbooks

Probability and Statistics

Read Explanation

Calculus

Read Explanation

Pure Maths

Read Explanation

Mechanics Maths

Read Explanation

Logic and Functions

Read Explanation

Applied Mathematics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.