Homework Help is Here – Start Your Trial Now!
Business
Economics
2.1.11. Theorem. If G is a simple graph, then diam G ≥ 3 ⇒ diam & ≤3. Proof: When diam G > 2, there exist nonadjacent vertices u, v = V(G) with no common neighbor. Hence every x = V (G) - {u, v} has at least one of (u, v) as a nonneighbor. This makes x adjacent in G to at least one of {u, v} in G. Since also uv E(G), for every pair x, y there is an x, y-path of length at most 3 in G through (u, v). Hence diam G≤ 3.

2.1.11. Theorem. If G is a simple graph, then diam G ≥ 3 ⇒ diam & ≤3. Proof: When diam G > 2, there exist nonadjacent vertices u, v = V(G) with no common neighbor. Hence every x = V (G) - {u, v} has at least one of (u, v) as a nonneighbor. This makes x adjacent in G to at least one of {u, v} in G. Since also uv E(G), for every pair x, y there is an x, y-path of length at most 3 in G through (u, v). Hence diam G≤ 3.

Principles of Microeconomics
Principles of Microeconomics
7th Edition
ISBN:9781305156050
Author:N. Gregory Mankiw
Publisher:N. Gregory Mankiw
Chapter22: Frontiers Of Microeconomics
Section: Chapter Questions
Problem 6PA
See similar textbooks
icon
Related questions
Question
Not use ai please
2.1.11. Theorem. If G is a simple graph, then diam G ≥ 3 ⇒ diam & ≤3. Proof: When diam G > 2, there exist nonadjacent vertices u, v = V(G) with no common neighbor. Hence every x = V (G) - {u, v} has at least one of (u, v) as a nonneighbor. This makes x adjacent in G to at least one of {u, v} in G. Since also uv E(G), for every pair x, y there is an x, y-path of length at most 3 in G through (u, v). Hence diam G≤ 3.
Transcribed Image Text:2.1.11. Theorem. If G is a simple graph, then diam G ≥ 3 ⇒ diam & ≤3. Proof: When diam G > 2, there exist nonadjacent vertices u, v = V(G) with no common neighbor. Hence every x = V (G) - {u, v} has at least one of (u, v) as a nonneighbor. This makes x adjacent in G to at least one of {u, v} in G. Since also uv E(G), for every pair x, y there is an x, y-path of length at most 3 in G through (u, v). Hence diam G≤ 3.
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Principles of Microeconomics
Principles of Microeconomics
Economics
ISBN:
9781305156050
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Microeconomic Theory
Microeconomic Theory
Economics
ISBN:
9781337517942
Author:
NICHOLSON
Publisher:
Cengage
Principles of Economics (MindTap Course List)
Principles of Economics (MindTap Course List)
Economics
ISBN:
9781305585126
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Principles of Microeconomics (MindTap Course List)
Principles of Microeconomics (MindTap Course List)
Economics
ISBN:
9781305971493
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
Principles of Economics, 7th Edition (MindTap Cou…
Principles of Economics, 7th Edition (MindTap Cou…
Economics
ISBN:
9781285165875
Author:
N. Gregory Mankiw
Publisher:
Cengage Learning
MACROECONOMICS FOR TODAY
MACROECONOMICS FOR TODAY
Economics
ISBN:
9781337613057
Author:
Tucker
Publisher:
CENGAGE L
SEE MORE TEXTBOOKS