Seven bridges Konigsberg aids in the investigation of DNA’s
Mathematics and biology have a synergistic relationship. Biology produces interesting problems, mathematics provides models to understand them, and biology returns to test the mathematical models. The Seven Bridges of Königsberg is a notable historical problem in mathematics. Its negative resolution by Leonhard Euler in 1735 laid the foundations of graph theory and presaged the idea of topology. The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time (one could not walk half way onto the bridge and then turn around and later cross the other half from the other side). Euler pointed out that the choice of route inside each landmass is irrelevant. The only important feature of a route is the sequence of bridges crossed. This allowed him to reformulate the problem in abstract terms (laying the foundations of graph theory), eliminating all features except the list of land masses and the bridges connecting them. In modern terms, one replaces each land mass with an abstract "vertex" or node, and each bridge with an abstract connection, an "edge", which only serves to record which pair of vertices (land masses) is connected by that bridge. The resulting mathematical structure is called a graph. Since only the connection information is relevant, the shape of pictorial representations of a graph may be distorted in any way without changing the graph itself. Only the existence (or lack) of an edge between each pair of nodes is significant. For example, it does not matter whether the edges drawn are straight or curved, or whether one node is to the left or right of another. Euler then formed an abstract model of the bridges and their adjoining land masses and founded the principles of Eulerian graphs on this model....
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