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Chapter 10: Problem 60

How many nonisomorphic simple graphs are there with six vertices and four edges?

Short Answer

Expert verified
There are 12 nonisomorphic simple graphs with six vertices and four edges.

Step by step solution

01

Understand Nonisomorphic Graphs

Nonisomorphic graphs are graphs that cannot be transformed into each other simply by renaming vertices. Two graphs are nonisomorphic if they have different structures.
02

Define Simple Graphs

Simple graphs are graphs without loops and multiple edges between the same pair of vertices. We are looking for such graphs with six vertices and exactly four edges.
03

Determine Possible Configurations

Analyze different ways of arranging six vertices with four edges without considering vertex labels. Each configuration must be unique in structure.
04

Generate Configurations

List all potential edge combinations and eliminate those that are isomorphic to each other. Consider the connectivity and structure of each graph.
05

Count Unique Configurations

After generating and analyzing the configurations, count the number of nonisomorphic (unique) structures.

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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 the study of graphs, which are mathematical structures used to model pairwise relations between objects. A graph is made up of vertices (also called nodes) and edges (connections between pairs of vertices). Graph theory helps us understand and solve problems related to networks, such as social networks, computer networks, and road maps. It is widely applied in computer science, biology, logistics, and more. One key aspect of graph theory is determining how different graphs can be classified and compared, often based on their structure and connectivity.
Simple Graphs
A simple graph is a type of graph that has no loops or multiple edges between the same pair of vertices. This means each connection (edge) is unique and directly links two different vertices. In the context of this exercise, we are dealing with simple graphs that have exactly six vertices and four edges. Simple graphs are easier to analyze compared to multigraphs, as their properties are more straightforward. They are fundamental in learning graph theory because they form the basis for understanding more complex graph types.
Isomorphism
Isomorphism in graph theory refers to the concept where two graphs can be considered the same if one can be transformed into the other simply by renaming vertices. In other words, isomorphic graphs have the same structure but may have different vertex labels. The process of checking isomorphism involves ensuring that there is a one-to-one correspondence between the vertices and edges of the two graphs. If such a mapping exists, the graphs are isomorphic. They share identical properties, such as the number of vertices, edges, and the arrangement of connections.
Nonisomorphic Graphs
Nonisomorphic graphs are graphs that cannot be transformed into each other by renaming vertices. They have different structures, meaning their connectivity patterns differ in ways that prevent them from being identical. In this exercise, when asked to find nonisomorphic simple graphs with six vertices and four edges, we need to identify all unique structural configurations. This involves systematically considering all possible ways to arrange four edges among six vertices and ensuring no two graphs in the set are isomorphic. This step is crucial for understanding the diversity of graph structures.
Graph Configurations
Graph configurations refer to the different ways of arranging edges among a given set of vertices. In our exercise, we look at configurations of six vertices connected by four edges. Each unique arrangement represents a different graph configuration. To find nonisomorphic simple graphs, we list all potential configurations, check for isomorphism, and eliminate duplicates. Through this method, we determine the number of unique (nonisomorphic) graph structures. This process helps in understanding how complex and varied graph structures can be, even with constraints on the number of vertices and edges.

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Most popular questions from this chapter

In an old puzzle attributed to Alcuin of York \((735-804),\) a farmer needs to carry a wolf, a goat, and a cabbage across a river. The farmer only has a small boat, which can carry the farmer and only one object (an animal or a vegetable). He can cross the river repeatedly. However, if the farmer is on the other shore, the wolf will eat the goat, and, similarly, the goat will eat the cabbage. We can describe each state by listing what is on each shore. For example, we can use the pair \((F G, W C)\) for the state where the farmer and goat are on the first shore and the wolf and cabbage are on the other shore. [The symbol \(\emptyset\) is used when nothing is on a shore, so that \((F W G C, \emptyset)\) is the initial state. \(]\) a) Find all allowable states of the puzzle, where neither the wolf and the goat nor the goat and the cabbage are left on the same shore without the farmer. b) Construct a graph such that each vertex of this graph represents an allowable state and the vertices representing two allowable states are connected by an edge if it is possible to move from one state to the other using one trip of the boat. c) Explain why finding a path from the vertex representing \((F W G C, \emptyset)\) to the vertex representing \((\emptyset, F W G C)\) solves the puzzle. d) Find two different solutions of the puzzle, each using seven crossings. e) Suppose that the farmer must pay a toll of one dollar whenever he crosses the river with an animal. Which solution of the puzzle should the farmer use to pay the least total toll?

Show that a simple graph \(G\) is bipartite if and only if it has no circuits with an odd number of edges.

Suppose that a connected planar simple graph with \(e\) edges and \(v\) vertices contains no simple circuits of length 4 or less. Show that \(e \leq(5 / 3) v-(10 / 3)\) if \(v \geq 4 .\)

Fleury's algorithm, published in \(1883,\) constructs Euler circuits by first choosing an arbitrary vertex of a connected multigraph, and then forming a circuit by choosing edges are successively. Once an edge is chosen, it is removed. Edges are chosen successively so that each edge begins where the last edge ends, and so that this edge is not a cut edge unless there is no alternative. Give a variant of Fleury's algorithm to produce Euler paths.

How many nonisomorphic simple graphs are there with \(n\) vertices, when \(n\) is $$\begin{array}{llll}{\text { a) } 2 ?} & {\text { b) } 3 ?} & {\text { c) } 4 ?}\end{array}$$
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