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Q1)

Constrcut the adjacency matrix and the adjacency lists for the graph G belowr.

x1

O

/ \

/ \

x2 O—–O x3

|\ /|

| \ / |

| / |

| / \ |

x4 |/ \| x0

O O

Graph G

Q2)

Constrcut the adjacency matrix and the adjacency lists for the graph G below,

where the weights associated with edges represent distances between nodes. If

no edge is present, it is equivalent to having a distance equal infinti.

x0 O

|\ Graph G

5 | \ 7

| \

| \

x1 O—-O x2

4 |\

| \7

5| \

| 4 \

x4 O—-O x3

Q3)

Consider the clique graph below.

x2 O—–O x3

|\ /|

| \ / |

| / |

| / \ |

x4 |/ \| x1

O—–O

Graph G

a) How many subgraphs of G with 3 nodes are there?

b) How many of the subgraphs defined in part(a) are induced subgraphs?

Q4) Consider a digraph D on 5 nodes, named x0, x1,.., x4, such that its

adjacency matrix contains 1’s in all the elements above the diagonal

A[0,0], A[1,1], A[2,2],.., etc, and contains 0’s in all the elements

along and below this diagonal.

| 0 1 1 1 1 |

| 0 0 1 1 1 |

| 0 0 0 1 1 |

| 0 0 0 0 1 |

| 0 0 0 0 0 |

a) Draw this digraph.

b) Form a table to record for each node its indegree, outdegree, and degree.

Q5) Determine whether the graphs G and H given below are isomorphic. If the

graphs are isomorphic find a 1-to-1 mapping between their nodes. Otherwise,

state why they are not.

x1 y1

O O

/ \ / \

/ \ / \

/ \ / \

x2 O O x3 y2 O—-/——-\—-O y3

| | | / \ |

| | | / \ |

| | | / \ |

| | |/ \|

x4 O——-O x5 y4 0 O y5

Graph G Graph H

Q6) Determine whether the graphs G and H given below are isomorphic. If the

graphs are isomorphic find a 1-to-1 mapping between their nodes. Otherwise,

state why they are not.

x1 y1

O O

/ \ /|\

/ \ / | \

x2 O—–O x3 y2 O–|–O y3

| /| | | |

| / | | | |

| / | | | |

| / | | | |

x4 |/ | x5 y4 O | O y5

O O \ | /

\ / \|/

\ / O

O y6

x6

Graph G Graph H

Q7) Determine whether the digraphs G and H given below are isomorphic. If they are,

find a 1-to-1 mapping between their nodes. Otherwise, explain why they are not.

x1 x3 y1 y2

O–<--o O---->—o

| / | |

| / | |

\|/ / | |

| / | |

|/ | |

| /|\ \|/

/| | |

/ | | |

/ | | |

|/_ | | |

/ | | |

x2 O–<--O x4 y3 O--->—-O y4

Digraph G Digraph H

Q8)

State whether the following arguement is True or False, and justify your

answer: If H is an induced subgraph of graph G, then the complement of H

must be an induced subgraph of the complement of G.

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