How to Draw a Multigraph with Degrees {1,1,4,5,6,7}

Math Problem Statement

It seems there was an issue generating the image. Let me describe how you can manually draw the multigraph based on the degrees { 1 , 1 , 4 , 5 , 6 , 7 } {1,1,4,5,6,7}.

Steps to draw the multigraph: Vertices: Draw 6 vertices, label them from 𝑉 1 V 1 to 𝑉 6 V 6 , and assign the following degrees:

𝑉 1 V 1 and 𝑉 2 V 2 each with degree 1. 𝑉 3 V 3 with degree 4. 𝑉 4 V 4 with degree 5. 𝑉 5 V 5 with degree 6. 𝑉 6 V 6 with degree 7. Edges: Start connecting the vertices:

Connect 𝑉 1 V 1 and 𝑉 6 V 6 with 1 edge. Connect 𝑉 2 V 2 and 𝑉 5 V 5 with 1 edge. Connect 𝑉 3 V 3 , 𝑉 4 V 4 , 𝑉 5 V 5 , and 𝑉 6 V 6 using multiple edges and loops: For example, 𝑉 6 V 6 can have several edges connected to 𝑉 4 V 4 , and a loop (edge connecting 𝑉 6 V 6 to itself) can be used to fulfill the degree requirement of 7. Loops and Multiple Edges: Since it's a multigraph, use additional edges between vertices or loops to meet the degree conditions where necessary.

Solution

To manually draw the multigraph based on the given degrees {1, 1, 4, 5, 6, 7}, follow these detailed steps:

Step 1: Draw the Vertices

Draw 6 vertices and label them $V_1$ through $V_6$ .
Assign the degrees as follows:
- $V_1$ and $V_2$ each with degree 1.
- $V_3$ with degree 4.
- $V_4$ with degree 5.
- $V_5$ with degree 6.
- $V_6$ with degree 7.

Step 2: Connect the Vertices

Connect $V_1$ and $V_6$ with 1 edge.
- $V_1$ now has a degree of 1 (fulfilled).
- $V_6$ now has a degree of 1, needing 6 more edges to fulfill its degree.
Connect $V_2$ and $V_5$ with 1 edge.
- $V_2$ now has a degree of 1 (fulfilled).
- $V_5$ now has a degree of 1, needing 5 more edges to fulfill its degree.

Step 3: Add More Edges and Loops

Connect $V_3$ to $V_4$ with 2 edges.
- $V_3$ now has a degree of 2, needing 2 more edges.
- $V_4$ now has a degree of 2, needing 3 more edges.
Connect $V_3$ to $V_6$ with 2 edges.
- $V_3$ now has a degree of 4 (fulfilled).
- $V_6$ now has a degree of 3, needing 4 more edges.
Connect $V_4$ to $V_5$ with 3 edges.
- $V_4$ now has a degree of 5 (fulfilled).
- $V_5$ now has a degree of 4, needing 2 more edges.
Connect $V_5$ to $V_6$ with 2 edges.
- $V_5$ now has a degree of 6 (fulfilled).
- $V_6$ now has a degree of 5, needing 2 more edges.
Add a loop to $V_6$ .
- $V_6$ now has a degree of 7 (fulfilled).

Step 4: Final Check

$V_1$ : Degree 1 (fulfilled).
$V_2$ : Degree 1 (fulfilled).
$V_3$ : Degree 4 (fulfilled).
$V_4$ : Degree 5 (fulfilled).
$V_5$ : Degree 6 (fulfilled).
$V_6$ : Degree 7 (fulfilled).

Your multigraph is now complete, with all vertices satisfying the required degrees.

Would you like a more detailed explanation of any of the steps? Here are some additional related questions:

What is the difference between a simple graph and a multigraph?
How are loops counted in the degree of a vertex?
Can you explain the concept of graph isomorphism?
What is the significance of the degree sequence in a graph?
How do Eulerian paths and circuits relate to the degrees of vertices in a graph?

Tip: When drawing multigraphs, remember that loops count as 2 towards the degree of the vertex they're connected to.

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Math Problem Analysis

Mathematical Concepts

Graph Theory
Multigraphs
Degree of Vertices

Formulas

Theorems

Handshaking Lemma

Suitable Grade Level

Grades 9-12

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