This show how tree vertexes are established and come together. this work with Breadth and in depth functioning

1. 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.
a b c d e
a 0 6 0 5 0
b 6 0 1 0 3
c 0 1 0 4 8
d 5 0 4 0 0
e 0 3 8 0 0
Describe the graph and why it is consistent with the matrix.
As we see here, there will be 5 vertices (nodes) and 8 edges (directional lines)
defining the paths from “a” to “e”. Since the matrices stipulates where the 5 nodes
will be, we are able to place 5 markers with the letter A → E into the doughnuts.
Now it is just a matter of assembling lines to the doughnuts using the matrix
provided. Going from left to right we can assess the line weight and where each line
meets up to each node.
A
D
E
B
C
6 5
1 34
8
0
0

2. How many simple paths are there from vertex 1 to vertex 5?
Explain. Which is the shortest of those paths?
a b c d e
a 0 6 0 5 0
b 6 0 1 0 3
c 0 1 0 4 8
d 5 0 4 0 0
e 0 3 8 0 0
Simple Paths
A → B → E = 3
A → D → E = 5
A → B → D → E = 6
A → B → C → E = 15
A →D → C → E = 17
A → D → B → C → E = 14
A → B → D → C → E = 18
A → B → C → D → E = 11
As we see in the simple paths, A → B → E = 3
is the shortest route from (A → E).

3. TreesINSTRUCTIONS
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,
The heads of finance and alumni relations answer to the vice president.
Each dean oversees three department chairpersons,
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?
This will fall under the hierarchical tree graph. I used these many a year when
making re-call rosters for our personnel. This is also known as an inverted tree.
The next slide will assemble this ensemble or orchestrated positions.

4. Hierarchy Tree
College President
Provost
Administrative
Assistant
Dean 1
Chair
Person 1
Department
Faculty
Chair
Person 2
Department
Faculty
Chair
Person 3
Department
Faculty
Dean 2
Chair
Person 4
Department
Faculty
Chair
Person 5
Department
Faculty
Chair
Person 6
Department
Faculty
Dean 3
Chair
Person 7
Department
Faculty
Chair
Person 8
Department
Faculty
Chair
Person 9
Department
Faculty
Vice President
Administrative
Assistant
Head of Finance
Head of Alumni
Relations
As we see, The tree is assembled and it is definitely an open tree of dissemination of
information through proper channels. If the College President and the Head of Alumni
circumvent the Vice President and Administrative Assistant on the information highway,
they are basically by passing two of the vertices within the dissemination process. It’s like
cutting out a branch and cloning the limb to the main branch. Therefore, “NO” this would
not continue to be a tree in this portion of the graph.