Let G = (V, E) be a graph (no loops, no multi-edges, undirected, as always. Let d be the smallest degrees of a vertex in G. Let D be the largest degree of a vertex in G. Let e = |E| and let n = |V|. Prove that
Let G = (V, E) be a graph (no loops, no multi-edges, undirected, as always. Let d be the smallest degrees of a vertex in G. Let D be the largest degree of a vertex in G. Let e = |E| and let n = |V|. Prove that
Chapter1: Equations, Inequalities, And Mathematical Modeling
Section1.1: Graphs Of Equations
Problem 6ECP: Use symmetry to sketch the graph of xy2=1.
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![**Title: Understanding Graph Degree Inequalities**
**Introduction:**
Let \( G = (V, E) \) be a graph, characterized by:
- No loops
- No multi-edges
- Undirected edges
**Definitions:**
- \( d \): The smallest degree of a vertex in \( G \).
- \( D \): The largest degree of a vertex in \( G \).
- \( e = |E| \): The number of edges in the graph.
- \( n = |V| \): The number of vertices in the graph.
**Theorem:**
Prove that the following inequality holds for the degrees and edges in graph \( G \):
\[
\frac{d}{2} \leq \frac{e}{n} \leq \frac{D}{2}
\]
This inequality showcases the relationship between the average degree of the vertices in the graph and the extremal degrees (smallest and largest) of vertices in the graph. Understanding and proving this inequality is a fundamental exercise in graph theory, providing insights into the structural properties and distributions of degrees in graphs.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F734f4932-9c15-4e07-80c9-d54d94f3ad88%2F4928f69f-ebe9-4c37-9210-ca5b4bfaacdf%2F6gs9j0a_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Understanding Graph Degree Inequalities**
**Introduction:**
Let \( G = (V, E) \) be a graph, characterized by:
- No loops
- No multi-edges
- Undirected edges
**Definitions:**
- \( d \): The smallest degree of a vertex in \( G \).
- \( D \): The largest degree of a vertex in \( G \).
- \( e = |E| \): The number of edges in the graph.
- \( n = |V| \): The number of vertices in the graph.
**Theorem:**
Prove that the following inequality holds for the degrees and edges in graph \( G \):
\[
\frac{d}{2} \leq \frac{e}{n} \leq \frac{D}{2}
\]
This inequality showcases the relationship between the average degree of the vertices in the graph and the extremal degrees (smallest and largest) of vertices in the graph. Understanding and proving this inequality is a fundamental exercise in graph theory, providing insights into the structural properties and distributions of degrees in graphs.
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