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Chapter 11: Problem 15

Draw all nonisomorphic simple graphs with four vertices.

Short Answer

Expert verified
There are 8 nonisomorphic simple graphs with four vertices: 1. G0: No edges (4 disconnected vertices) 2. G1: One edge (two vertices connected and two disconnected vertices) 3. G2: Two edges forming two disconnected pairs of connected vertices 4. G3: Two edges forming a path of three vertices and one disconnected vertex 5. G4: Three edges connecting all vertices to a central vertex 6. G5: Four edges forming a triangle and an additional edge from one of the triangle vertices to the fourth vertex 7. G6: Four edges forming a cycle with all four vertices 8. G7: Five edges connecting every vertex to every other vertex (a complete graph)

Step by step solution

01

Draw graphs with 0 edges

There is only one simple graph with 0 edges: four disconnected vertices, labeled A, B, C, and D. We'll call this graph G0.
02

Draw graphs with 1 edge

There's only one simple graph with 1 edge: connect any two vertices (e.g., A to B) and leave the other two disconnected. We'll call this graph G1.
03

Draw graphs with 2 edges

For two edges, we have two possibilities: 1. Connecting two pairs of vertices (A-B and C-D): Call this graph G2. 2. Connecting three vertices in a line or a path (A-B, B-C): Call this graph G3.
04

Draw graphs with 3 edges

For three edges, we have two possibilities: 1. Connecting all three vertices to one central vertex (A-B, A-C, and A-D): Call this graph G4. 2. Forming a triangle and connect the fourth vertex to one vertex (A-B, B-C, C-A, A-D): Call this graph G5.
05

Draw graphs with 4 edges

There's only one simple graph with 4 edges: forming a cycle with all four vertices (A-B, B-C, C-D, D-A). Call this graph G6.
06

Draw graphs with 5 edges

For a simple graph with 4 vertices and 5 edges, we'd need to connect every vertex to every other vertex (A-B, B-C, C-D, D-A, A-C). Call this graph G7.
07

Check for isomorphisms

We need to compare the drawn graphs and check if any of them are isomorphic. If yes, we remove one of the isomorphic graphs from our list. In this case, none of the drawn graphs are isomorphic, so we have the following nonisomorphic simple graphs with four vertices: 1. G0: No edges (4 disconnected vertices) 2. G1: One edge (two vertices connected and two disconnected vertices) 3. G2: Two edges forming two disconnected pairs of connected vertices 4. G3: Two edges forming a path of three vertices and one disconnected vertex 5. G4: Three edges connecting all vertices to a central vertex 6. G5: Four edges forming a triangle and an additional edge from one of the triangle vertices to the fourth vertex 7. G6: Four edges forming a cycle with all four vertices 8. G7: Five edges connecting every vertex to every other vertex (a complete graph) These are all the nonisomorphic simple graphs with four vertices.

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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 branch of mathematics that focuses on the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph is composed of vertices (also called nodes or points) and edges (links or lines) that connect them. It's a powerful tool used in various fields such as computer science, biology, and social science to solve problems involving networks, like social networks or the internet.

Graphs can be classified in various ways according to their properties, such as being simple, connected, or complete. Understanding these classifications is crucial for solving problems like determining the number of nonisomorphic simple graphs for a given number of vertices, which illustrates the flexibility and application of graph theory concepts in combinatorial problems.
Isomorphism

What Is Graph Isomorphism?

Isomorphism in graph theory refers to a kind of 'sameness' between two graphs. Specifically, two graphs are considered isomorphic if there's a way to re-label the vertices of one graph to obtain the other graph. This implies that isomorphic graphs have the same number of vertices and edges and the same connectivity structure.

To determine isomorphism between graphs, we systematically compare the structure of graphs. If there's a one-to-one correspondence between their vertex sets that preserves adjacency and non-adjacency, the graphs are isomorphic. Recognizing nonisomorphic graphs, like in the given exercise, is essential for understanding different possible configurations in a network.
Simple Graph

Defining a Simple Graph

A simple graph is a graph that doesn't contain any loops (edges connected at both ends to the same vertex) or multiple edges (more than one edge between the same pair of vertices). It's the most basic form of a graph used in graph theory.

Simple graphs are often used to represent undirected relationships, such as friendships in a social network. They are also the starting point for more complex graph types, including those with directional edges (digraphs) or weighted edges (where each edge has a value). In the provided exercise, all graphs are simple graphs, showcasing the different ways just four vertices can be connected (or not connected) without creating loops or parallel edges.
Connected Graph

Connected Graph Essentials

In graph theory, a connected graph is one where there is a path from any vertex to any other vertex. It means that you can get from any one point to any other point without jumping out of the graph. A connected graph allows for uninterrupted movement through the vertices, representing a continuous network.

The concept of a connected graph is vital when we need to ensure all elements in a network are accessible from each other, such as in communication networks. Not all the graphs from the exercise are connected: G0, G1, G2, and G3 are examples of disconnected graphs as they have vertices that cannot be reached from others.
Complete Graph

The Concept of a Complete Graph

A complete graph, denoted by the symbol Kn where n is the number of vertices, is a simple graph in which every pair of distinct vertices is connected by a unique edge. In other words, it's a graph where every vertex has an edge to every other vertex.

The concept of a complete graph is important in problems that require all possible connections between elements without redundancy, like in scheduling or circuit design. The final nonisomorphic simple graph from the exercise, G7, is an example of a complete graph for four vertices, commonly denoted as K4.

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