,MAT3707 Assignment 3 (COMPLETE ANSWERS)
2024 (156556) - DUE 30 August 2024 ; 100%
TRUSTED Complete, trusted solutions and
explanations.
QUESTION 1 Consider the following two graphs: a b d c e f G1 u
v w y x z G2 Are G1 and G2 isomorphic? A. Yes, because they
both have the same degree sequence. B. Yes, because function
g with g(a) = u, g(b) = v, g(c) = w, g(d) = x, g(e) = y, and g(f) = z is
an isomorphism. C. Yes, because function h with h(a) = z, h(b) =
w, h(c) = x, h(d) = y, h(e) = u, and h(f) = v is an isomorphism. D.
Yes, because they both have the same number of vertices, the
same number of edges and are both connected. E. No, they are
not isomorphic.
To determine if the graphs G1G_1G1 and G2G_2G2 are
isomorphic, you need to verify several conditions:
1. Number of Vertices: Both graphs must have the same
number of vertices.
2. Number of Edges: Both graphs must have the same
number of edges.
3. Degree Sequence: The degree of each vertex (the number
of edges connected to it) must match between
corresponding vertices.
4. Connectivity: The graphs must have the same overall
structure (i.e., if two vertices are connected in one graph,
, the corresponding vertices should be connected in the
other graph).
Options Analysis:
Option A: "Yes, because they both have the same degree
sequence."
The degree sequence is important, but it alone does not
guarantee isomorphism. Even if two graphs have the same
degree sequence, they may have different structures. This
option is not necessarily correct.
Option B: "Yes, because function ggg with g(a)=ug(a) = ug(a)=u,
g(b)=vg(b) = vg(b)=v, g(c)=wg(c) = wg(c)=w, g(d)=xg(d) = xg(d)=x,
g(e)=yg(e) = yg(e)=y, and g(f)=zg(f) = zg(f)=z is an isomorphism."
This option suggests a specific mapping. If this mapping
preserves the adjacency structure of the graphs, then
G1G_1G1 and G2G_2G2 are isomorphic. However,
whether this mapping is correct depends on checking the
structure, which we cannot confirm without the graph
diagrams.
Option C: "Yes, because function hhh with h(a)=zh(a) = zh(a)=z,
h(b)=wh(b) = wh(b)=w, h(c)=xh(c) = xh(c)=x, h(d)=yh(d) =
yh(d)=y, h(e)=uh(e) = uh(e)=u, and h(f)=vh(f) = vh(f)=v is an
isomorphism."
Similarly, this option suggests a different mapping. Again,
confirming this would require checking the graph
structure.
, Option D: "Yes, because they both have the same number of
vertices, the same number of edges, and are both connected."
While these conditions are necessary for isomorphism,
they are not sufficient on their own. The exact adjacency
structure must still be preserved.
Option E: "No, they are not isomorphic."
This would be correct if the mappings in options B and C
fail to preserve adjacency, or if the structure is different in
some way.
Conclusion:
Without the actual graph diagrams, determining the correct
answer is challenging. However, typically, if one of the
mappings suggested (in options B or C) preserves the adjacency
structure, then the graphs are isomorphic. Given the
information presented, Option B or Option C could be correct if
the mappings are indeed isomorphisms.
QUESTION 2 Which of the following is a necessary condition for
a graph to be bipartite? A. The graph has no odd cycles. B. The
graph has even cycles. C. The graph has an Euler cycle. D. The
graph is planar. E. The graph has a Hamiltonian
circuit.QUESTION 3 In a connected graph, if every vertex has an
2024 (156556) - DUE 30 August 2024 ; 100%
TRUSTED Complete, trusted solutions and
explanations.
QUESTION 1 Consider the following two graphs: a b d c e f G1 u
v w y x z G2 Are G1 and G2 isomorphic? A. Yes, because they
both have the same degree sequence. B. Yes, because function
g with g(a) = u, g(b) = v, g(c) = w, g(d) = x, g(e) = y, and g(f) = z is
an isomorphism. C. Yes, because function h with h(a) = z, h(b) =
w, h(c) = x, h(d) = y, h(e) = u, and h(f) = v is an isomorphism. D.
Yes, because they both have the same number of vertices, the
same number of edges and are both connected. E. No, they are
not isomorphic.
To determine if the graphs G1G_1G1 and G2G_2G2 are
isomorphic, you need to verify several conditions:
1. Number of Vertices: Both graphs must have the same
number of vertices.
2. Number of Edges: Both graphs must have the same
number of edges.
3. Degree Sequence: The degree of each vertex (the number
of edges connected to it) must match between
corresponding vertices.
4. Connectivity: The graphs must have the same overall
structure (i.e., if two vertices are connected in one graph,
, the corresponding vertices should be connected in the
other graph).
Options Analysis:
Option A: "Yes, because they both have the same degree
sequence."
The degree sequence is important, but it alone does not
guarantee isomorphism. Even if two graphs have the same
degree sequence, they may have different structures. This
option is not necessarily correct.
Option B: "Yes, because function ggg with g(a)=ug(a) = ug(a)=u,
g(b)=vg(b) = vg(b)=v, g(c)=wg(c) = wg(c)=w, g(d)=xg(d) = xg(d)=x,
g(e)=yg(e) = yg(e)=y, and g(f)=zg(f) = zg(f)=z is an isomorphism."
This option suggests a specific mapping. If this mapping
preserves the adjacency structure of the graphs, then
G1G_1G1 and G2G_2G2 are isomorphic. However,
whether this mapping is correct depends on checking the
structure, which we cannot confirm without the graph
diagrams.
Option C: "Yes, because function hhh with h(a)=zh(a) = zh(a)=z,
h(b)=wh(b) = wh(b)=w, h(c)=xh(c) = xh(c)=x, h(d)=yh(d) =
yh(d)=y, h(e)=uh(e) = uh(e)=u, and h(f)=vh(f) = vh(f)=v is an
isomorphism."
Similarly, this option suggests a different mapping. Again,
confirming this would require checking the graph
structure.
, Option D: "Yes, because they both have the same number of
vertices, the same number of edges, and are both connected."
While these conditions are necessary for isomorphism,
they are not sufficient on their own. The exact adjacency
structure must still be preserved.
Option E: "No, they are not isomorphic."
This would be correct if the mappings in options B and C
fail to preserve adjacency, or if the structure is different in
some way.
Conclusion:
Without the actual graph diagrams, determining the correct
answer is challenging. However, typically, if one of the
mappings suggested (in options B or C) preserves the adjacency
structure, then the graphs are isomorphic. Given the
information presented, Option B or Option C could be correct if
the mappings are indeed isomorphisms.
QUESTION 2 Which of the following is a necessary condition for
a graph to be bipartite? A. The graph has no odd cycles. B. The
graph has even cycles. C. The graph has an Euler cycle. D. The
graph is planar. E. The graph has a Hamiltonian
circuit.QUESTION 3 In a connected graph, if every vertex has an
