Prove that the following conditions on a connected graph Γ are equivalent.Γ is a tree.Given any two vertices v and w in Γ, there is a unique reduced edge path from v to w.For every edge e # E(Γ), removing e from Γ disconnects the graph. (Note: Removing e does not remove its associated vertices.)If G is finite then #V (Γ) = #E(Γ) + 1.

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Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Prove that the following conditions on a connected graph Γ are equivalent.

Γ is a tree.
Given any two vertices v and w in Γ, there is a unique reduced edge path from v to w.
For every edge e # E(Γ), removing e from Γ disconnects the graph. (Note: Removing e does not remove its associated vertices.)
If G is finite then #V (Γ) = #E(Γ) + 1.

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