Problem 14
Question
A connected, planar graph contains 10 vertices and divides the plane into seven regions. Compute the number of edges in the graph.
Step-by-Step Solution
Verified Answer
The connected, planar graph with 10 vertices and dividing the plane into 7 regions has 15 edges.
1Step 1: Use Euler's formula for connected planar graphs
We have Euler's formula: V - E + F = 2. We are given V = 10 and F = 7. Our goal is to find E.
2Step 2: Substitute the known values into Euler's formula
We can plug the values of V and F into Euler's formula: 10 - E + 7 = 2
3Step 3: Solve the equation for E
Now, let's solve the equation to find the value of E:
10 - E + 7 = 2
17 - E = 2
- E = 2 - 17
- E = -15
Now, multiply both sides by -1 to get the value of E:
E = 15
4Step 4: State the final answer
The connected, planar graph with 10 vertices and dividing the plane into 7 regions has 15 edges.
Key Concepts
Understanding Connected Planar GraphsEuler's Formula and Computing Number of EdgesVertices and Faces in Graphs
Understanding Connected Planar Graphs
A connected planar graph is a particular type of graph that can be drawn on a plane without any of the edges crossing each other. The fact that the graph is connected means there is a path between any two vertices, ensuring the graph is a single piece. In simpler words, imagine it like a map where you can travel from any city to any other without having to cross an ocean.
Connected planar graphs are bounded by some fascinating mathematical properties. One of the intriguing aspects is how they divide the plane into regions, which we call faces. The number of faces is related to the number of vertices (points where lines meet) and edges (lines connecting vertices) through a famous relationship known as Euler's formula.
Connected planar graphs are bounded by some fascinating mathematical properties. One of the intriguing aspects is how they divide the plane into regions, which we call faces. The number of faces is related to the number of vertices (points where lines meet) and edges (lines connecting vertices) through a famous relationship known as Euler's formula.
Euler's Formula and Computing Number of Edges
Euler's formula is an essential cornerstone in graph theory. It provides a powerful relationship between an undirected graph's vertices (V), edges (E), and faces (F), in one simple equation: \( V - E + F = 2 \). This relationship is specific to connected planar graphs.
To apply this formula effectively, remember that you'll often have two of these components known and need to solve for the third. To find the number of edges, you would rearrange Euler's equation to \( E = V + F - 2 \). For instance, in our example with 10 vertices and 7 faces, having this equation makes computing the number of edges straightforward. You plug in the numbers to get \( E = 10 + 7 - 2 = 15 \) edges, allowing students to grasp the unseen structure of such graphs with ease.
To apply this formula effectively, remember that you'll often have two of these components known and need to solve for the third. To find the number of edges, you would rearrange Euler's equation to \( E = V + F - 2 \). For instance, in our example with 10 vertices and 7 faces, having this equation makes computing the number of edges straightforward. You plug in the numbers to get \( E = 10 + 7 - 2 = 15 \) edges, allowing students to grasp the unseen structure of such graphs with ease.
Vertices and Faces in Graphs
The beauty of graph theory is in how vertices and faces combine to tell a story about the graph's layout. The vertices are the 'hubs' of activity, where lines intersect, and faces are the 'spacious areas' surrounded by edges. They visually represent the regions, like countries on a map.
When we discuss a connected planar graph, the number of faces correlates with the complexity of its structure. For example, a graph that creates more regions typically involves more edges, which can become complicated. Each face is created by a cycle of edges, and additional vertices may be needed to support the expanding network. Understanding this interaction between vertices, edges, and faces can make complex graphs feel less intimidating and more like a geometrical puzzle waiting to be solved.
When we discuss a connected planar graph, the number of faces correlates with the complexity of its structure. For example, a graph that creates more regions typically involves more edges, which can become complicated. Each face is created by a cycle of edges, and additional vertices may be needed to support the expanding network. Understanding this interaction between vertices, edges, and faces can make complex graphs feel less intimidating and more like a geometrical puzzle waiting to be solved.
Other exercises in this chapter
Problem 12
Draw the graph with the given adjacency matrix. $$\begin{aligned}&\qquad a\begin{array}{lllll}& b &c & d \end{array} \\\ &\begin{array}{lllll}a \\ b \\ c \\ d \
View solution Problem 13
The adjacency matrix of a simple graph has the form $$A=\left[\begin{array}{l|l} A_{1} & 0 \\ \hline 0 & A_{2} \end{array}\right]$$ What can you say about the g
View solution Problem 14
Find the chromatic number of each map or graph. Wheel graph \(W_{n}\)
View solution Problem 15
A connected, planar graph contains 24 edges. It divides the plane into 13 regions. How many vertices does the graph have?
View solution