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Let n be a positive integer. Show that every connected graph on n vertices has at least n – 1 - edges.

Let n be a positive integer. Show that every connected graph on n vertices has at least n – 1 - edges.

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
Problem 80EQ
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Let n be a positive integer. Show that every connected graph onn vertices has at least n – 1 edges. Hint: use the following two lemmas proved in the Lecture Notes. Lemmal : (this is Lemma 3.4 from the Lecture Notes) Let G be a connected graph. An edge e is not a cut-edge if and only if it belongs to a cycle. Lemma2 : (this follows directly from Theorem 3.4 from the Lecture Notes) For every positive integer n, every tree on n vertices has n – 1 edges. -
Transcribed Image Text:Let n be a positive integer. Show that every connected graph onn vertices has at least n – 1 edges. Hint: use the following two lemmas proved in the Lecture Notes. Lemmal : (this is Lemma 3.4 from the Lecture Notes) Let G be a connected graph. An edge e is not a cut-edge if and only if it belongs to a cycle. Lemma2 : (this follows directly from Theorem 3.4 from the Lecture Notes) For every positive integer n, every tree on n vertices has n – 1 edges. -
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