Question

Exercise 2 Given the following graph: a. Write the formal description of the graph, G=(V,E) b. Show the Adjacency Matrix representation C. Show the Adjacency List representation d. Calculate step by step the shortest paths from a e. Show the DFS tree/forest from a f. Show the BFS tree/forest from a g MST using Prim h. MST using Kruskal

Answer 1

According to HomeworkLib policy we answer first Four Sub parts., but I answer 5 for u.

a) In a Graph if there is a Weighted edge between vertexes a and vertex b then Put weight value of a to b in adjacency Matrix otherwise we take in the adjacency matrix

b) Adjacency Matrix: In a Graph if there is a direct edge between vertexes a and vertex b then Put weight value of a to b in adjacency Matrix otherwise we take in the adjacency matrix

Adjacency Matrix Representation of given Weighted graph is as follows

	a	b	c	d	e	f	g	h
a	0	2	3
b	2	0		2
c	3		0	1
d		2	1	0	2	4
e				2	0	1	2
f				4	1	0	2	1
g					2	2	0	3
h						1	3	0

c) Adjacency List Representation:

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d) Dijkstra’s shortest-path algorithm to compute the shortest path from “a” to all other vertices of a graph is as follows

Step D(d) P(d) D(e) Ple) D(f) P(f) D(g) P(8) D(h) P(h) Dla) P(a) D(b) P(b) D(c) P(c) 2,a(min) 3,a 2, a 3,a 2,a 1 | 88 4(2+2),b 4(3+1),c 3,a 8 8 8 8 8 2,a 3,a 4,0 2,a 2,a 3, a 3,a 6(4+2),d 6,0 6,0 6,0 6,0 8(4+4),d 7(6+1),e 7,e 8(6+2), e 8,e 4,c 8(7+1),f 8(7+1),f 6 2,a 3,a 4,c 7,e 2,a 3,a 7,e 8,e

Step 0: Initially we take a as a minimum Weight and highlight that value

Step 1: In this we consider the minimum weight other than Vertex a that is at Vertex b we have minimum weight so highlight that value

Step 2: In this we consider the minimum weight other than Vertex a & b that is at Vertex c we have minimum weight so highlight that value

Step 3: In this we consider the minimum weight other than Vertex a,b & c that is at Vertex d we have minimum weight so highlight that value

Step 4: In this we consider the minimum weight other than Vertex a,b,c & d that is at Vertex e we have minimum weight so highlight that value

Step 5: In this we consider the minimum weight other than Vertex a,b,c,d & e that is at Vertex f we have minimum weight so highlight that value

Step 6: In this we consider the minimum weight other than Vertex a,b,c,d,e & f that is at Vertex g we have minimum weight so highlight that value

Step 7: In this we consider the minimum weight other than Vertex a,b,c,d,e,f & g that is at Vertex h we have minimum weight so highlight that value

The shortest path tree from the above routing table.

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h) Kruskal’s Algorithm to find the Minimum Cost Spanning Tree (MCST) of a graph G as follows:

Here consider those edges in each step which are minimum and doesnot form Loops

Step 1:                                                                 

Step 2:                                               

مسن

Step 3:

Step 4:                                                                 

Step 5:                                               

Step 6:

Step 7:   

For n- Vertices we get (n-1) Edges in MCST. Here in graph having 8 Vertices so we get 7 Edges

Which is Required MCST of a graph