Researchers study social media for all kinds of reasons, and graph theory is behind much of that research. Since networks are everywhere, graph theory is everywhere, too. Graph theory is used to model and study all kinds of things that affect our daily lives: from transatlantic shipping routes to integrated circuits, from molecular bonds to animal food webs.
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Link copied to clipboard. Share Look Out! Big Ideas are Hiding in Your Arithmetic. Share The Value of Mathematical Simplicity. Share Making Math Relevant. Share Pictures Help us Remember Math. In this context a graph or network as many people use the terms interchangeable consists of:. Some of you may be reading this document via the Web. That is also a graph, with each document file being a node and each hypertext link the thing you click on to go elsewhere an arc.
Graph theory has a relatively long history in classical mathematics. In Euler solved the problem of whether, given the map below of the city of Konigsberg in Germany, someone could make a complete tour, crossing over all 7 bridges over the river Pregel, and return to their starting point without crossing any bridge more than once. See if you can reach the same conclusion as Euler did. The picture below shows the city, but simplified so that just the river and bridges are shown - do you think that someone could make a complete tour, crossing over all 7 bridges over the river, and return to their starting point without crossing any bridge more than once.
In this note we consider two graph theory problems relating to spanning trees and shortest paths in detail and outline the multinational tax planning problem. More about graph theory can be found here. Consider the following problem. In the diagram shown below we have four wells in an offshore oilfield nodes 1 to 4 below and an on-shore terminal node 5 below. The four wells in this field must be connected together via a pipeline network to the on-shore terminal. The various pipelines that can be constructed are shown as links in the diagram below and the cost of each pipeline is given next to each link.
What pipelines would you recommend be built? It is clear that this particular problem is a specific example of a more general problem - namely given a graph such as that shown above which links would we use so that:. This problem is called the shortest spanning tree SST problem. Note here that the phrase "minimal spanning tree" MST is also often used instead of the phrase "shortest spanning tree" SST.
For example, in the diagram above, one possible structure connecting all the points together would consist of the links , , , and , but there are other structures where the total cost of the links used is smaller than in this structure e.
Note here that the Kruskal algorithm only applies to graphs in which all the links are undirected. For the graph shown above, applying the Kruskal algorithm and starting with the shortest least cost link, we have:. Hence the SST of the graph shown above consists of the links , , and total cost 14 and is as shown below. Note here that in applying the Kruskal algorithm we often encounter the situation in which two or more links have the same cost e.
In such a case the order in which these equal cost links are considered does not matter [different SST's may result depending upon the order chosen but they will all have the same total cost and this cost will be the minimum that can be achieved]. Verbal Ability. Interview Questions. Company Questions. Artificial Intelligence. Cloud Computing. Data Science. Angular 7. Machine Learning.
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Ethical Hacking. Computer Graphics. Software Engineering. Web Technology. Cyber Security. Have vertices representing the start node, each intermediate layer which gets two: one per direction of travel , and one outgoing node for the signals that are simply lost on the other side.
Now the result you want can be calculated by performing a simple calculation on the weighted adjacency matrix. This is an absurdly general concept, and so applications of graph theory will pop up in all kinds of neat places.
Alexander Schrijver, A Course in Combinatorial Optimization includes multiple applications just search for "Application" in the text. The parts that appear to get used the most are shortest paths, maximum flows, maximum matchings and arc-disjoint paths. Of course, it is usually the algorithms that get applied rather than the existence theorems, but this is common to almost all applications of mathematics, and most of the theorems have algorithmic proofs which, in some cases, are the standard proofs.
I won't spoil the punchline, but let me just mention that both the maximum-flow and the minimum-cut problem appeared in the middle of the 20th century out of non-mathematical considerations, and both were intended to be applied to the same network. I generally have the impression is that you don't often see "mere" graphs in the applied world i.
So you will often see networks, matrices and various other models instead. Note that what is commonly called a "sparse matrix" is essentially a graph with numbers on its edges; so every sparse matrix algorithm that is used in the real world is an application of graph theory.
Graph theory, like many fields of mathematics, can provide a more precise way of describing what people in the real world are already doing. For example, a colleague and I are investigating how library catalogers over the years have, at least since the mid 19th century, created graph structures within library catalogs - in their book, index card, and database record forms. Fascinating questions - both abstract and practical - arise. A nice example is the "Kevin Bacon number" and similar problems originally the Erdos number or the shortest path to Mr X.
Sign up to join this community. The best answers are voted up and rise to the top. Stack Overflow for Teams — Collaborate and share knowledge with a private group. Create a free Team What is Teams? Learn more. Graph Theory Applications? Ask Question. Asked 8 years, 9 months ago. Active 1 year ago. Viewed 17k times. Cause I wonder what applications this have on the real world.
Does it solve certain problems and stuff?
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