Networks are frequently referred to as graphs in mathematics, and graph theory is the branch of mathematics that deals with the study of graphs. Vertices are the points used to depict interconnected items, while edges are the connections between them. Graph TheoryĪ graph is a visual representation of a collection of things where some object pairs are linked together. However, although it might not sound very applicable, there are actually an abundance of useful and important applications of graph theory. There are many more interesting areas to consider and the list is increasing all the time graph theory is an active area of mathematical research.Graph theory might sound like an intimidating and abstract topic. 5.S: Graph Theory (Summary) Hopefully this chapter has given you some sense for the wide variety of graph theory topics as well as why these studies are interesting.5.9.3: Transportation Networks and Flows.5.7: Weighted Graphs and Dijkstra's Algorithm.Our goal in this activity is to discover some criterion for when a bipartite graph has a matching. 5.6: Matching in Bipartite Graphs Given a bipartite graph, a matching is a subset of the edges for which every vertex belongs to exactly one of the edges.Our goal is to find a quick way to check whether a graph (or multigraph) has an Euler path or circuit. An Euler circuit is an Euler path which starts and stops at the same vertex. 5.5: Euler Paths and Circuits An Euler path, in a graph or multigraph, is a walk through the graph which uses every edge exactly once.5.4: Coloring Given any map of countries, states, counties, etc., how many colors are needed to color each region on the map so that neighboring regions are colored differently? How is this related to graph theory? Well, if we place a vertex in the center of each region (say in the capital of each state) and then connect two vertices if their states share a border, we get a graph.Notice that the definition of planar includes the phrase “it is possible to.” This means that even if a graph does not look like it is planar, it still might be. 5.3: Planar Graphs When is it possible to draw a graph so that none of the edges cross? If this is possible, we say the graph is planar (since you can draw it on the plane).We want our definition to be precise and unambiguous, but it also must agree with our intuition for the objects we are studying. Is there a graph with no edges? We have to look at the definition to see if this is possible. The definition is the agreed upon starting point from which all truths in mathematics proceed. Crafting good definitions is not easy, but it is incredibly important. 5.2: Definitions The way we avoid ambiguities in mathematics is to provide concrete and rigorous definitions.When two vertices are connected by an edge, we say they are adjacent. Graphs are made up of a collection of dots called vertices and lines connecting those dots called edges. 5.1: Prelude to Graph Theory Pictures like the dot and line drawing are called graphs. Since then it has blossomed in to a powerful tool used in nearly every branch of science and is currently an active area of mathematics research. Graph Theory is a relatively new area of mathematics, first studied by the super famous mathematician Leonhard Euler in 1735.
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