What is Graph Theory like? ( After a few weeks of study anyway. ) Let's take the The Konigsberg Bridges problem as an example. This problem marks the start of Graph Theory and was solved by Leonhard Euler. The question was to find a route through the city crossing each bridge only once. Then the Graph Theorist enters the scene.

He translates the relevant parts of the city to a graph such that the seven bridges become seven edges and the various city areas shrink to mere vertices. He translates the original question to a question about the graph: is this graph Eulerian? If so, then the answer to the original question is yes.

But what exactly is an Eulerian graph? An Eulerian graph is a connected graph which contains an Eulerian trail. A graph theorists knows a theorem which helps him to decide if the graph contains an Eulerian trail. We however, have to explain first the words connected graph and Eulerian trail. An Eulerian trail is a closed trail that includes every edge. Again. Two new words! Closed trail and edge ( = connection between two vertices. ) A closed trail is a trail with start and finish at the same vertex ( = a dot in a graph. ) A trail is a walk in which all edges, but not necessarily all vertices, are different. A walk of length k is a succession of k edges of the form uv, vw, wx, ..., yz. This walk is denoted by uvwx...yz, and is referred to as a walk between u and z. Remains the word connected graph. A graph is connected if there is a path between each pair of vertices.

Why not show all these definitions, relations and corresponding theorems in a graph?! Exactly. ( There is an aspect of self-reference here but I lack the knowledge of terms to exactly describe it. Maybe after M381. )

I created this graph using Personal Brain. It is possible to share and publish brains on the internet in order to collaborate on some project. MT365 is the first course I am using Personal Brain for. It is too soon to evaluate though.

1-2017 More on the randomness of randomness.

4 weeks ago

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