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Interactive Audio Lesson
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Understanding Graphs and Vertices
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Today, we will explore a simple graph with 6 vertices. Can anyone explain what we mean by a 'simple graph'?
A simple graph has no loops or multiple edges between the same pair of vertices.
Correct! The graph consists only of edges connecting distinct vertices. Now, what happens when we delete a vertex?
The edges connected to that vertex are also removed.
Exactly! This is a key concept we'll use. Deleting a vertex decreases the total edge count based on that vertex's degree. Remember this phrase: 'Deleting a vertex, reduces the edges pertaining to its degree'.
Deletion of Vertices and Edge Count
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Now, let's discuss specific vertex deletion. If I delete vertex v1, I end up with 7 edges remaining. What does this imply?
It means that the degree of vertex v1 must have been 1, because 8 total edges minus 1 is 7.
Great observation! The relationship here is that the original edge count decreases by the vertex degree. Each vertex deletion gives us an equation to work with. Let's write this down!
So we can keep track of the degree of each vertex like this?
Precisely! Keep a log of the degrees as we formulate our equations.
Using Logical Reasoning to Find Edge Count
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Let's summarize the equations we've formed from our vertex deletions. We derived that each deletion correlates to a specific degree.
And we can add all those equations together to obtain a single equation?
Exactly! By summing all results, we derive a larger equation involving the total edge count, which is our goal.
What happens next once we have that big equation?
From there, we apply the handshaking theorem to relate the sum of degrees to the cardinality of edges, allowing us to solve for the original edge count.
Introduction & Overview
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Quick Overview
Standard
The section provides a thorough investigation into a simple graph with 6 vertices, analyzing how deleting various vertices affects the remaining number of edges. By leveraging logical rules and properties of graphs, it arrives at a conclusion about the cardinality of the original graph's edge set, emphasizing the relationship between vertex degree and edge count.
Detailed
Detailed Summary
In this section, we explore an unknown simple graph G with 6 vertices and analyze how deleting various vertices affects the edge set. When a vertex is removed, the number of edges in the graph decreases by the degree of that vertex. Given specific degrees for each vertex associated with the number of edges that remain when each vertex is deleted, we can derive the total edge count for the original graph G. The problem emphasizes using logical reasoning over brute-force methods to efficiently determine the cardinality of the edge set. Through this analysis, we utilize the handshaking theorem, which connects the degree of vertices to the total number of edges in a graph, reinforcing the relationship that the sum of the degrees of all vertices equals twice the total number of edges.
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Understanding the Problem
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In question number 2 you are given an unknown simple graph G, the graph G is not known to you known to you in the sense that you are just given that it has 6 vertices but exact cardinality of edge set is not given. But it is given to you that your graph G is such that if you delete the vertex v from the graph then you are left with 7 edges. If you delete the vertex v from the graph you are left with 6 edges and so on.
Detailed Explanation
In this part of the question, we have a graph G with 6 vertices, but we don't know how many edges it has. We do know something interesting about the edges: if we remove a vertex from the graph, the number of edges changes according to the degree of that vertex. Specifically, the graph loses edges equivalent to the number of edges connected to the removed vertex. This indicates we need to use the properties of graph theory to deduce the total number of edges in G.
Examples & Analogies
Imagine a community of 6 friends where each friend has a certain number of connections (friendships) with each other. If one friend (vertex) leaves the community, the remaining friendships (edges) decrease. Just as we can count the total number of friendships knowing how many each friend has, we can find the total edges in graph G by considering how many edges disappear when each friend leaves.
Using Vertex Degrees
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So, if that is the case the question asks you to find out the cardinality of the edge set of the original graph. Again do not try to do a brute force and try all possible graphs in your mind and then hit upon the answer because that will take enormous amount of time. Instead we will try to apply some rules of logic and properties of graph here.
Detailed Explanation
Instead of guessing and checking, we will systematically analyze the problem using the relationship between vertex deletion and edge count. The principle states that removing a vertex from a graph decreases the total edge count by the degree of that vertex. Therefore, we can set up an equation based on the given conditions of the graph after removing individual vertices, allowing us to calculate the total number of edges in G without testing every possible graph.
Examples & Analogies
Imagine you’re organizing a group event and need to calculate how many connections (friendships) are left if certain friends (vertices) are not attending. Rather than randomly guessing connections for everyone, you can make a chart of who is connected to whom. This would allow you to easily determine how many friends remain and their interactions without unnecessary guesswork.
Setting Up Equations
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So that means what I can say is that my graph G is such that the edge set cardinality minus the degree of v is 7 because it is given that after deleting vertex v in the graph you are left with 7 edges. That means in the leftover graph which I obtain after deleting v if I would have added the edges which were incident with vertex v and how many such edges would have been there? : degree of v number of edges that would have given me the cardinality of the edge set.
Detailed Explanation
Based on the properties outlined, we create equations for each vertex in G. For each vertex v_i removed, the resulting edge count is related to its degree. For instance, if removing vertex v_1 leaves us with 7 edges, we can express this mathematically: (total edges in G) - (degree of v_1) = 7. This process is repeated for all vertices with their respective edge counts after removal.
Examples & Analogies
Think of this as needing to account for how many friends are left after each invited friend declines the invitation to a party. By counting who has said they will attend after each decline, you can amend your final guest count more accurately without needing to recalculate the entire group each time.
Summation and Solving for Edges
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Now if I sum all these 6 equations I get that 6 times the cardinality of E minus the summation of the degree of 6 vertices in the graph is 36. And now I can apply the handshaking theorem which says that if you take the summation of the degrees of all the vertices in your original graph it is same as twice the number of edges.
Detailed Explanation
After establishing the equations, we sum them together to form a compact representation of edge counts, leading to a single central equation. We observe that the sum of vertex degrees equals twice the total number of edges (known as the handshaking theorem). Thus, we can resolve for the total edge count from this finetuned equation.
Examples & Analogies
Similar to creating a budget for a project involving team members. As you account for each member’s contributions, you gather all their input into one sheet, allowing you to calculate the total effort put in. Just as the budget figures can be combined, here we can combine equations to determine the total edges.
Conclusion of Edge Set Cardinality
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So, now if I take the summation of the degrees of all the vertices in the original graph it is same as twice the number of edges. So, now I have 1 equation just involving the unknown which is my cardinality of the edge set. So, I get my edge set’s cardinality to be 9.
Detailed Explanation
Applying the handshaking theorem, we relate our summation of degrees to twice the edge count, yielding a conclusive equation exclusively in terms of the total edge number. Solving this final equation provides the edge set's cardinality, establishing a clear result: the graph has 9 edges.
Examples & Analogies
This final step is akin to reaching the conclusion of your budget calculations, leading you to the final project cost. By analyzing each contributor's sum of efforts, you can coherently determine the total required resource, finishing your calculations just as we finalize the edge count.
Definitions & Key Concepts
Learn essential terms and foundational ideas that form the basis of the topic.
Key Concepts
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Vertex Deletion: The process of removing a vertex from the graph along with its connected edges.
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Cardinality of Edge Set: The total number of edges within a graph.
Examples & Real-Life Applications
See how the concepts apply in real-world scenarios to understand their practical implications.
Examples
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If vertex v1 has a degree of 1, deleting it reduces the edge count from 8 to 7.
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If vertex v2 has a degree of 2, deleting it reduces the edge count based on its attached edges.
Memory Aids
Use mnemonics, acronyms, or visual cues to help remember key information more easily.
🎵 Rhymes Time
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When you take a pt away, edges drop by its degree all day!
📖 Fascinating Stories
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Imagine a village with 6 houses connected by paths. Removing a house takes away its paths, changing how people can travel.
🧠 Other Memory Gems
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Remember "DROPS": Deleting Reduces Overall Paths and Connections Sum (Edges).
🎯 Super Acronyms
GRADED
- Graph Removal Affects Degree Edge Decrease.
Flash Cards
Review key concepts with flashcards.
Glossary of Terms
Review the Definitions for terms.
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Term: Simple Graph
Definition:
A graph with no loops or multiple edges between the same pair of vertices.
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Term: Degree of a Vertex
Definition:
The number of edges incident to a vertex.