Question 4 Let G(V, E) be a simple graph with minimum degree 8(G) ≥ 1. The line graph of G, denoted by L(G), has vertex set V₂ = {e: e E E). Two vertices i,j E V, are adjacent if and only if the corresponding edges i,j EE are incident to a common vertex. Example: (a) a (c) b C 9 d Graph H a g b Graph G Graph L(G) Draw the line graphs L(H) and L(K) of graphs H and K respectively. Graph K (b) Let G be a simple r-regular graph with r≥2. Is (G) also a regular graph? If L(G) Is regular graph, what is the common degree in it? Characterise, with justification, all simple graphs G such that G and its line graph C(G) are isomorphic. That is, describe, with justification, all simple graphs G such that G and L(G) are isomorphic.
Question 4 Let G(V, E) be a simple graph with minimum degree 8(G) ≥ 1. The line graph of G, denoted by L(G), has vertex set V₂ = {e: e E E). Two vertices i,j E V, are adjacent if and only if the corresponding edges i,j EE are incident to a common vertex. Example: (a) a (c) b C 9 d Graph H a g b Graph G Graph L(G) Draw the line graphs L(H) and L(K) of graphs H and K respectively. Graph K (b) Let G be a simple r-regular graph with r≥2. Is (G) also a regular graph? If L(G) Is regular graph, what is the common degree in it? Characterise, with justification, all simple graphs G such that G and its line graph C(G) are isomorphic. That is, describe, with justification, all simple graphs G such that G and L(G) are isomorphic.
Holt Mcdougal Larson Pre-algebra: Student Edition 2012
1st Edition
ISBN:9780547587776
Author:HOLT MCDOUGAL
Publisher:HOLT MCDOUGAL
Chapter12: Angle Relationships And Transformations
Section12.5: Reflections And Symmetry
Problem 20E
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Transcribed Image Text:Question 4
Let G(V, E) be a simple graph with minimum degree (G) ≥ 1. The line graph of G, denoted
by L(G), has vertex set V₂ = {e: e E E). Two vertices i,j € V, are adjacent if and only if the
corresponding edges i,j EE are incident to a common vertex.
Example:
(a)
(c)
a
(d)
b
C
G
d
Graph H
a
g
b
Graph G
Graph L(G)
Draw the line graphs L(H) and L(K) of graphs H and K respectively.
(b)
Let G be a simple r-regular graph with r≥2. Is C(G) also a regular graph? If C(G) Is
regular graph, what is the common degree in it?
Graph K
Characterise, with justification, all simple graphs G such that G and its line graph C(G)
are isomorphic. That is, describe, with justification, all simple graphs G such that G and
L(G) are isomorphic.
Show that no line graph can contain the star K₁3 as an induced subgraph.
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