4. Let T be a tree with n ≥ 1 vertices. (a) Show that T has a vertex of degree 1.

Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter3: Matrices
Section3.7: Applications
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4. Let T be a tree with n ≥ 1 vertices.
(a) Show that I has a vertex of degree 1.
(b) Show that T has n - 1 edges.
(c) Let G be a connected graph and suppose that
e = {x,y} is an edge of G lying on a cycle.
Show that G' = (V, E \ {e}) is connected.
(d) Conclude that every connected graph on n
vertices has a spanning tree.
Transcribed Image Text:4. Let T be a tree with n ≥ 1 vertices. (a) Show that I has a vertex of degree 1. (b) Show that T has n - 1 edges. (c) Let G be a connected graph and suppose that e = {x,y} is an edge of G lying on a cycle. Show that G' = (V, E \ {e}) is connected. (d) Conclude that every connected graph on n vertices has a spanning tree.
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