Apply improved Dijkstra’s algorithm to the following graph to find the shortest path from A to every other vertex in the graph. (b) Apply Kruskal’s algorithm to the graph to find a minimum spanning tree.

a) Illustrate the sequence of vertices of this graph visited using depth-first search traversal starting at vertex g. b) Illustrate adjacency list representation and adjacency matrix representation, respectively, for this graph. What are the advantages and disadvantages of those two representations? c) Describe an algorithm to find in the graph a path illustrated below that goes through every edge exactly once in each direction.

(c) Use Kruskal's algorithm to find a minimal spanning tree for the following graph, explaining each step. Draw the minimal spanning tree. Give the total weight of the spanning tree. 3 D B 2 6 4 E 5 4 F с 9 6 3 A

(b) Write down the degree of cach vertex in the following graph. Hence, verify the Handshaking Theorem.

A) Give the minimum spanning tree using Prim’s algorithm (starting at vertex A). B) Give the minimum spanning tree using Kruskal’s algorithm. C) What is the total weight of the minimum spanning tree?

a) write the algorithm  b) Briefly explain why the algorithm solves the problem, and the correctness c ) State and briefly explain the running time of your algorithm

A) Give the depth-first traversal of the graph (). B) Give the breadth-first traversal of the graph.

Graph Theory Question 9.6 Graph G is an n-vertex graph. For integer n ≥ 3, let every new graph formed by deleting a vertex from G be a tree. Answer the following two questions: 1. What is the number of edges of G 2. Prove why G is a cycle of Cn on n vertices

18. Show that in a simple graph with at least two vertices there must be two vertices that have the same degree.

(3) All parts of this question refer to the following graph: A abi В Mon nas (i) How many bridges are there in the graph? D sessab (ii) How many cycles are there in the graph? Do not count cycles as different if they use exactly the sity 202 isity G (iii) How many different spanning trees does the graph contain? (iv) How many walks of length 5 start at I and end at L? same set of edges. For example, EFLKE and LFEKL count as the same cycle. H I (v) How many paths of length 8 start at A and end at D? (vi) Draw the spanning tree that results from a breadth first search, starting at A and using the de tas ask. standard alphabetical order for vertices. Your answer must have vertex labels on it. Also, for your tutor's convenience, please position the vertices similarly to how they are positioned above. Monash onas Sable task, ble task, task, sessable tas ask, y 20% sity 20213 ersity 2021. iversity 2021. Gersity 202 ssable sessableota OM essanle sessabi ask. asabre task. Monash able…