Let n nodes be given labeled 1, 2, ... n and define the triangular graph on these nodes to be the graph T_n in which there is an edge (i,j) if and only if i + j ≤ n + 1 and i ≠ j. The term "triangular" refers to the fact that the adjacency matrix has a triangular pattern. The adjacency lists representation is:

1: 2, 3, 4, 5
2: 1, 3, 4
3: 1, 2
4: 1, 2
5: 1

1. 1. Draw T₅
   2. Draw the breadth first search tree in T₅ that results when starting at node 1.
   3. Draw the breadth first search tree in T₅ that results when starting at node 5.
   4. Draw the depth first search tree in T₅ that results when starting at node 1.
   5. Draw the depth first search tree in T₅ that results when starting at node 5.

2. 1. Show the shortest path tree that results from Dijkstra's algorithm in T₅ when starting at node 1.
   2. Show the shortest path tree that results from Dijkstra's algorithm in T₅ when starting at node 5.

3. 1. Show the minimal spanning tree that results from Prim's algorithm in T₅ when starting at node 1.
   2. Show the minimal spanning tree that results from Prim's algorithm in T₅ when starting at node 5.
   3. Show the minimal spanning tree that results from Kruskal's algorithm in T₅.

4. True or false ( prove or give counterexample): Every triangular graph has a unique minimum spanning tree.

5. True or false ( prove or give counterexample): A connected edge-weighted undirected graph in which the edge weightes are distinct (no two weights are equal) has a unique minimum spanning tree.

6. Dijkstra's algorithm may be viewed as breadth first search with the queue of nodes to be visited replaced by a priority queue. In particular for a weighted connected graph with all the weights equal, Dijkstra's algorithm and BFS construct exactly the same search tree. Construct a weighted connected graph such that the spanning tree constructed by Dijkstra's algorithm and the tree by depth first search are the same.

7. Chapter 4, Exercise 9 (Bottleneck).