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"An Application of Graph Theory." Application of Graph Theory. N.p., n.d. Web. 01 May 25.
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Daddel, Ali A. "Dominance-directed Graph." Dominance-directed Graph. N.p., n.d. Web. 01 May 25.
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Professor Jen-Mei Chang, Ph.D: Introduction to Linear Algebra Group Quiz 2
The adjacency matrix represents one-way paths connecting flights.
From group quiz 2
This represents the directed graph of the airplane traffic control.
When squaring the adjacency matrix, it shows the number of two-way paths between two vertices.
Vertex: A point in a graph. This point can represent various things.
Edge: A link from one vertex to another.
A graph is called directed if its edges are directed, meaning that they have a direction.
A graph is called dominance-directed if for any pair of distinct vertices, u and v, either u v
or v u, but not both.
In addition to our examples, other real life applications of graph theory and linear algebra include circuits, image reconstructions, search engines, shopping websites, social network analysis, and more.
Adjacency Matrix: A matrix that represents which vertices are adjacent (connected) to each other.
This represents the number of two-way paths from city A to city B.
Introduced by Leonhard Euler, graph theory is a branch of mathematics which created a solution consisting the representation of a problem through a “graph”. The situation was to start from one point, travel through all edges without crossing any edge twice, and returning to the starting point.
Given a direct dominance graph, we have created an adjacency matrix that represents the performance of each team up to at most two rounds which is represented by the adjacency matrix, denoted by M, and the adjacency matrix squared, denoted by M2. Let’s call M2+ M matrix A. The team with the highest row sum in matrix A has the highest ranking. This ranking indicates that they are the best team in the competition, and thus they have the best chance of winning the tournament. The reason why we can’t go further in multiplying adjacency matrix by itself more than twice is because by a theorem relating to a dominance-directed graph, a vertex is allowed up to at most a 2-way path to another vertex.
Given the directed graph, we have created an adjacency matrix that represents one-way paths between two cities. The entries of the matrix indicates the number of one-way paths from different cities. This is the same when we raise the adjacency matrix to nth power. The entries in the new matrix indicate the number of n-way paths there are from different cities.
B
A
C
We can transform any type of graph into an adjacency matrix where we can study it by raising it to an nth power and adding it. The information we receive from the new matrices do not represent a new graph, rather it still applies back to the original graph. The information from these graphs show us how many n-way paths can be constructed, and which team has the best chance of winning a tournament. However, if the graph has limitations, then our computations and results have limitations as well. Our dominance-directed graph is inaccurate since we are assuming that a team either has a 100% chance or no chance of winning the tournament.
When cubing the adjacency matrix, it shows the number of three-way paths between two vertices.
This graph represents the number of three-way paths from city B to city D.
from Kakao
Team 4 has the highest rank out of all teams and the highest chance of winning the tournament.
Yay!
from Kakao