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.

1) Describe the graph and why it is consistent with the matrix.
2) How many simple paths are there from vertex 1 to vertex 5? Explain.Which is the shortest of those paths?
Part II: Trees

1) 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.
2) 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?

The graph with a given adjacency matrix is constructed. The number of simple paths connecting two concrete vertices in this graph is found. The shortest path among such paths is found. The tree that models the structure of a college staff is given.

Can anyone finish up this proof by continuing my preliminery work? I started but can't finish this. I know starting by adding up the point z is correct way, but just can't continue to show if and only if.
(See attached file for full problem description)
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Assume , , with rank (A) = m are given. Two different basic

What does it mean for two graphs to be the same? Let G and H be graphs. We say that G is isomorphic to H provided that there is a bijection f:V(G) -> V(H) so that for all a, b, in V(G) there is an edge connecting a and b (in G) if and only if there is an edge connecting f(a) and f(b) (in H). The function f is called an isomorphi

One of the problems of storing data in a matrix (a two-dimensional Cartesian structure) is that if not all of the elements are used, there might be quite a waste of space. In order to handle this, we can use a construct called a "sparse matrix", where only the active elements appear. Each such element is accompanied by its two i

AVL trees are a good implementation of binary search trees. Show (step by step) the AVL trees formed by inserting the numbers 3, 11, 2, 9, 8, 12, 10, 5, 4, 7, 6, 1, 13.

Please see attached questions.
This is three questions.
Question #1 - find the chromatic number of the graph.
Question #2 - It might be supposed that if a graph has a large number of vertices and
each vertex has a large degree, then the chromatic number would have to be large. Show
that this conjecture is incorrect by co

Please see the attached files for the fully formatted problems.
1. Given the equation below, find f(x) where y = f(x).
8y(6x - 7) - 12x(4y + 3) + 265 - 5(3x - y + 2) = 0.
2. Solve these linear equations for x, y, and z.
3x + 5y - 2z = 20; 4x - 10y -z = -25; x + y -z = 5
3. The value of y in Question 2 lies in the ran

I am posting one problem from Exercise 2.2, I need answer for 2.9.
I am posting another question from Exercise 3.1:
Problem 3.2.) Prove that a graph G is a forest if and only if every induced subgraphs of G contains a vertex of degree at most 1.
Problem 3.1) Draw all forests of order 6.
See attached file for full pr