How Does Vertex Degree Range Affect Edge Count in a Graph?

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In summary, Graph Theory is a branch of mathematics that deals with the study of graphs and their relationships between objects. A proof in Graph Theory is a logical and rigorous argument that uses definitions, axioms, and previously proven theorems. The first step in approaching a Graph Theory problem is to understand and break down the problem into smaller parts. Some common techniques used in Graph Theory proofs include induction, contradiction, and direct proof. To improve your skills in Graph Theory, it is important to have a strong foundation in basic mathematical concepts and regularly practice solving problems. Staying updated on current research and attending conferences and workshops can also enhance understanding of Graph Theory.
  • #1
SpatialVacancy
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Help me with a proof!

Suppose that [tex]G[/tex] is a graph with [tex]v[/tex] vertices and [tex]e[/tex] edges and that the degree of each vertex is at least [tex]d_{min}[/tex] and at most [tex]d_{max}[/tex]. Show that:



[tex]\dfrac{1}{2}d_{min}\ \cdot\ v \ \leq \ e \ \leq \ \dfrac{1}{2}d_{max}\ \cdot\ v [/tex]


I don't have an idea of where to start on this problem. Thank you for your help!
 
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Every edge connects 2 vertices. Suppose every edge had degree exactly dmin. How many edges would there be?
 
  • #3


Sure, I'd be happy to help you with this proof! Let's start by defining some terms to make things clearer. The degree of a vertex in a graph is the number of edges that are incident to that vertex. So for this problem, we have a graph G with v vertices and e edges, and the degree of each vertex is at least d_{min} and at most d_{max}.

To prove this statement, we will use the Handshaking Lemma, which states that the sum of the degrees of all vertices in a graph is equal to twice the number of edges. In other words,

\sum_{v \in V} deg(v) = 2e

where V is the set of all vertices in the graph.

Now, let's consider the minimum and maximum degrees of the vertices in G. We know that the degree of each vertex is at least d_{min} and at most d_{max}, so we can write the following inequalities:

d_{min} \leq deg(v) \leq d_{max}

for all vertices v in G. Multiplying both sides by v and summing over all vertices in G, we get:

\sum_{v \in V} d_{min} \leq \sum_{v \in V} deg(v) \leq \sum_{v \in V} d_{max}

Using the Handshaking Lemma, we can rewrite the left and right sides as:

d_{min}v \leq 2e \leq d_{max}v

Finally, dividing both sides by 2, we get the desired result:

\dfrac{1}{2}d_{min}\ \cdot\ v \ \leq \ e \ \leq \ \dfrac{1}{2}d_{max}\ \cdot\ v

Therefore, we have proven that the number of edges e in G must be between \dfrac{1}{2}d_{min}\ \cdot\ v and \dfrac{1}{2}d_{max}\ \cdot\ v, as desired. I hope this helps! Let me know if you have any other questions or if you need further clarification.
 

Related to How Does Vertex Degree Range Affect Edge Count in a Graph?

1. What is Graph Theory?

Graph Theory is a branch of mathematics that deals with the study of graphs, which are mathematical structures used to model relationships between objects. It is a fundamental tool in fields such as computer science, engineering, and social sciences.

2. What is a proof in Graph Theory?

A proof in Graph Theory is a logical and rigorous argument that uses definitions, axioms, and previously proven theorems to establish the truth of a statement about graphs. It is an essential part of the scientific process and ensures the validity of mathematical results.

3. How do I approach a Graph Theory problem?

The first step in approaching a Graph Theory problem is to understand the given problem and identify the relevant concepts and definitions. Then, try to break down the problem into smaller parts and use known theorems and techniques to solve each part. Finally, check your solution for accuracy and completeness.

4. What are some common techniques used in Graph Theory proofs?

Some common techniques used in Graph Theory proofs include induction, contradiction, and direct proof. Induction is often used to prove statements about graphs with a recursive structure. Contradiction is used to show that a statement cannot be false. Direct proof involves providing a logical argument using definitions and theorems to establish the truth of a statement.

5. How can I improve my skills in Graph Theory?

To improve your skills in Graph Theory, it is essential to have a strong foundation in basic mathematical concepts such as logic, set theory, and proof techniques. It is also helpful to regularly practice solving problems and studying different types of graphs and their properties. Additionally, staying updated on current research and attending conferences and workshops can also enhance your understanding of Graph Theory.

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