True or False? Justify your answer. Let G be a graph. 1. If G has no isolated vertex then G is connected. 2. If G has a path passing through all of its vertices then (G) ≥ 2. 3. If G has a cycle passing through all of its vertices then (G) ≥ 2. 4. If a(G) = 1 and G has more than one vertex then G is a complete graph.
True or False? Justify your answer. Let G be a graph. 1. If G has no isolated vertex then G is connected. 2. If G has a path passing through all of its vertices then (G) ≥ 2. 3. If G has a cycle passing through all of its vertices then (G) ≥ 2. 4. If a(G) = 1 and G has more than one vertex then G is a complete graph.
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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Exercise 1 only
Transcribed Image Text:Exercise
True or False? Justify your answer.
Let G be a graph.
1.
If G has no isolated vertex then G is connected.
2. If G has a path passing through all of its vertices then
(G) ≥ 2.
3. If G has a cycle passing through all of its vertices then
(G) ≥ 2.
4. If a(G) = 1 and G has more than one vertex then G is a complete graph.
Exercise 2.
Let G be a graph and let a and b E G such that d(a) = A(G) and d(b) = S(G) 2 1. Suppose
that b is the unique vertex in G such that d(b) = S(G). Define the following two sets:
A = {ve G: d(v) < d(u) Vue N(v)}
B = {ve G: d(v) > d(u) u € N(v)}
(1) Show that a A.
(2) Show that A is a stable set.
(3) Suppose that G is a bipartite graph such that G = G(A, B). Show that G contains
an even cycle.
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