Task Background: Graphs and Trees are useful in visualizing data and the relations within and between data sets. Conversely, it is also important to be able to represent graphs as databases or arrays, so that programs for processing the data can be written. 

Part I: Adjacency Matrix and Shortest Path 

Construct a graph based on the adjacency matrix that appears below. Label all nodes with indices consistent with the placement of numbers within the matrix. 

⌈ 0 6 0 5 0⌉ 

| 6 0 1 0 3 | 

| 0 1 0 4 8 | 

| 5 0 4 0 0 | 

⌊ 0 3 8 0 0⌋ 

Describe the graph and why it is consistent with the matrix. 

How many simple paths are there from vertex 1 to vertex 5? Explain. 

Which is the shortest of those paths? 

Part II: Trees 

Construct and describe a tree that indicates the following: 

A college president has 2 employees who answer directly to him or her, namely a vice president and provost. The vice president and provost each have an administrative assistant. Three deans answer to the provost, and the heads of finance and alumni relations answer to the vice president. Each dean oversees three department chairpersons, and each department chair oversees several faculty in each of their respective departments. 

Suppose that the professional correspondences are the same as above, with the addition that there is also a direct working relationship between the college president and the head of alumni relations (it is not necessary to draw this). Would the graph still be a tree? Why or why not?