Using the algorithm of your choice, determine one possible minimum-weight spanning tree

One can manually count path lengths in a graph using adjacency matrices. Using the simple example below, produces the following adjacency matrix: A B A 1 1 B 1 0 This matrix means that given two vertices A and B in the graph above, there is a connection from A back to itself, and a two-way connection from A to B. To count the number of paths of length one, or direct connections in the graph, all one must do is count the number of 1s in the graph, three in this case, represented in letter notation as AA, AB, and BA. AA means that the connection starts and ends at A, AB means it starts at A and ends at B, and so on. However, counting the number of two-hop paths is a little more involved. The possibilities are AAA, ABA, and BAB, AAB, and BAA, making a total of five 2-hop paths. The 3-hop paths starting from A would be AAAA, AAAB, AABA, ABAA, and ABAB. Starting from B, the 3-hop paths are BAAA, BAAB, and BABA. Altogether, that would be eight 3-hop paths within this graph. Write a program…

Given the following undirected unweighted graph G(V, E). a C What is E? b d e

Draw a graph G containing the following vertices and edges. V(G) = {0, 1, 2, 3, 4, 5, 6} E(G) = { (0, 1), (0, 2), (0, 3), (2, 1), (2, 4), (5, 1), (5, 3)}

Consider eight points on the Cartesian two-dimensional x-y plane. a g C For each pair of vertices u and v, the weight of edge uv is the Euclidean (Pythagorean) distance between those two points. For example, dist(a, h) : V4? + 1? = /17 and dist(a, b) = v2? + 0² = 2. Because many pairs of points have identical distances (e.g. dist(h, c) V5), the above diagram has more than one minimum-weight spanning tree. dist(h, b) = dist(h, f) Determine the total number of minimum-weight spanning trees that exist in the above diagram. Clearly justify your answer.

Given an n-sided regular polygon n, return the total sum of internal angles (in degrees). Examples sumPolygon (3) → 180 sumPolygon (4) → 360 sumPolygon (6) → 720

Given a Triangle of three vertices A(2,3), B(4,2), C(3,0), Apply the shearing to the Triangle with SH=4

Show the formal notation V(...) and E(...) of the graph that is described by the following adjacency matrix. List the edges with vertices in uppercase as follows (X,Y) A BCD 1 10 1 0 0 0 0 0 1 O 1 0 o 10 O 0 0 00 0 O 0 0 0 0 0 0 0 0 1 0 0 O 0 0 0 1 0 E G A. 1 B 1 D 1 V={ E={

Consider the following edge-weighted digraph with 8 vertices and 13 edges. V->W weight A->E 1 A->B B->C C->F 4 C->D D->G F->E 39 23 F->B 26 F->A 16 G->F G->C 40 34 H->D 95 H->G 17 Here is a graphical representation of the same edge-weighted digraph: (A) -3---->(B) -39-->(D) 16 95 (E)- -23- (F). 40 (G) (H) Suppose that you run the Bellman-Ford algorithm to compute the shortest paths from H to every other vertex. What is the distTo[] array immediately after the end of three passes of the algorithm (pass 1, 2, and 3)? Each pass consists of relaxing the 13 edges in the order given above. Here is the distTo[] array before the beginning of pass 1: v A B CDE F H. distTo(v] Answer Your answer should be a sequence of 8 integers, separated by whitespace,

Submit solutions to the following problems in a single pdf document preferably completed using LaTeX. It is likely that you will need to use programming (Python) or a matrix calculator to solve most of the problems. You do not need to submit the code, but you should include explanations of all your conclusions. 3. Consider the cat polygon with vertices at (0, 0), (3, 3), (3, 6), (2, 7), (1, 6), (-1, 6), (-2, 7), (-3, 6), and (-3, 3). a. Write a single matrix that will make the cat's face twice as tall, half as wide, and rotate anti- clockwise 45 degrees. What is the determinant of the transformation matrix? b. Calculate the area of the cat's face prior to transformation and after the transformation. How does the change in area relate to the determinant of the transformation matrix?