(d) Suppose that a graph G is regular of degree r, where r is odd. (i) Prove that G has an even number of vertices. (ii) Prove that the numof G is a multiple of r.
(d) Suppose that a graph G is regular of degree r, where r is odd. (i) Prove that G has an even number of vertices. (ii) Prove that the numof G is a multiple of r.
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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Help with no (d) please

Transcribed Image Text:(a) Write down the chromatic number of each of the graphs below. Justify your answer by
colouring the graph.
(i) Cube.
(ii) Krst
(iii) Wn
01 120
1000 1
(b) A graph G has adjacency matrix A =
10011
20.100
0 1 10 0
(i) Is Ga simple graph?
(i) Write down the degree sequence for G.
(c) Prove by induction that the number of edges of K, is 01-1).
2
(d) Suppose that a graph G is regular of degree r, where r is odd.
(i) Prove that G has an even number of vertices.
(ii) Prove that the numof G is a multiple of r.
(e) A simple graph has 20 vertices. Any two distinct vertices u and v are such that
deg(u) + deg(v) 2 20. Prove by contradiction that the graph is connected.
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