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(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
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 

(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. Scientific WorkPlace
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. Scientific WorkPlace
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