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What is the Degree of a Vertex in a Graph?
Grade Level:
Class 10
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The 'degree' of a vertex (or node) in a graph is simply the number of edges connected to that vertex. Think of it as how many 'friends' a particular point has in a network.
Simple Example
Quick Example
Imagine a group of friends in your class. If Rahul is friends with Priya, Sameer, and Anjali, then Rahul's 'degree' in this friend network is 3, because he has 3 connections. Each friendship is an 'edge' from Rahul.
Worked Example
Step-by-Step
Let's consider a simple graph representing bus routes in a small town. The 'vertices' are bus stops (A, B, C, D) and 'edges' are direct routes between them.
Graph Description:
- Stop A has direct routes to Stop B and Stop C.
- Stop B has direct routes to Stop A and Stop D.
- Stop C has a direct route to Stop A.
- Stop D has a direct route to Stop B.
Step 1: Identify Vertex A. Count the edges connected to A. Edges are A-B and A-C. --- Step 2: So, Vertex A has 2 edges connected to it. Its degree is 2. --- Step 3: Identify Vertex B. Count the edges connected to B. Edges are B-A and B-D. --- Step 4: So, Vertex B has 2 edges connected to it. Its degree is 2. --- Step 5: Identify Vertex C. Count the edges connected to C. The only edge is C-A. --- Step 6: So, Vertex C has 1 edge connected to it. Its degree is 1. --- Step 7: Identify Vertex D. Count the edges connected to D. The only edge is D-B. --- Step 8: So, Vertex D has 1 edge connected to it. Its degree is 1.
Answer: Degree of A = 2, Degree of B = 2, Degree of C = 1, Degree of D = 1.
Why It Matters
Understanding vertex degrees helps us analyze networks, which are everywhere! From social media connections to city traffic flow, knowing how connected each point is can help in fields like AI, data science, and even planning efficient logistics. You could be a data scientist analyzing how information spreads!
Common Mistakes
MISTAKE: Counting each edge twice (once for each vertex it connects) when calculating the degree of a single vertex. | CORRECTION: For a specific vertex, only count the edges that directly 'touch' or 'come out of' that vertex once.
MISTAKE: Confusing the number of vertices in a graph with the degree of a specific vertex. | CORRECTION: The degree is about connections to ONE vertex, not the total number of points in the graph.
MISTAKE: Ignoring loops (an edge connecting a vertex to itself) or multi-edges (multiple edges between the same two vertices) when they are present. | CORRECTION: A loop counts as two edges towards the degree of that vertex, and each multi-edge counts as a separate connection.
Practice Questions
Try It Yourself
QUESTION: In a graph, Vertex P is connected to Vertex Q, Vertex R, and Vertex S. What is the degree of Vertex P? | ANSWER: 3
QUESTION: A graph has vertices X, Y, Z. X is connected to Y and Z. Y is connected to X. Z is connected to X. What is the degree of Vertex X? | ANSWER: 2
QUESTION: Consider a graph with vertices A, B, C, D. A is connected to B. B is connected to C. C is connected to D. A is also connected to C. What is the degree of Vertex C? | ANSWER: 3
MCQ
Quick Quiz
What is the degree of a vertex that has only one edge connected to it?
1
2
3
The Correct Answer Is:
B
The degree of a vertex is the count of edges connected to it. If only one edge is connected, its degree is 1.
Real World Connection
In the Real World
Think about your favourite social media app like Instagram or WhatsApp. Each person is a 'vertex', and if you follow someone or are in a group chat with them, that's an 'edge'. The 'degree' of your profile would be how many people you are connected to. Companies use this to understand popular users or how trends spread!
Key Vocabulary
Key Terms
VERTEX: A point or node in a graph | EDGE: A line connecting two vertices | GRAPH: A collection of vertices and edges | NETWORK: Another name for a graph, often used in real-world contexts
What's Next
What to Learn Next
Now that you understand vertex degrees, you can explore 'paths and cycles' in graphs. This will help you understand how to navigate through a network, like finding the shortest route on Google Maps, which builds directly on knowing how many connections each point has.
