8) Solve the all-pairs shortest paths problem on the following directed graph using a dynamic-programming algorithm. All steps of the solution are written in the details. 6 8 -5

(a) Writeup Prove Theorem 1 Using Theorem 1, describe an algorithm in words to check bipartiteness and analyze its run time.

Algorithm : (1. Single-destination Shortest Path, 2. Bellman-Ford, 3. Negative-Weight Cycles), Dynamic Programming Define and prove a recurrence for the following problem: Given a directed graph G = (V, E) with edge weight function w : E → R and a source vertex s ∈ V , find a shortest path from s to v for every vertex v ∈ V .

[Problem 10] Run the Dijkstra's algorithm on the following directed graph and find all shortest paths between vertex A and all others. B A 7 4 1 D E F

Q.3 (a) Find the degree sequence of the graphs: K6,5, W, (b) Find All values of n such that the graphs: (i) K. (i) C are bipartite (c) Draw the graph with the given incidence matrix 1 100 1011 1 1 100 00 0 01 1001 0 0011 10 0 00 00 0ol Thope the solution is clear and the line is understandable

1) Find the adjacency matrix M for the following graph (see attached) 1a) Find M^2,M^3,M^4. What do the entries represent? 1b)Express the transitive closure of a general graph G in terms of its adjacency matrix.

6) Solve the single-source shortest-paths problem for the following graph using the Bellman-Ford algorithm. Let the vertex s the single-source. All steps of the solution are written in the details. 5 2 6 4 7