Gephi

I just discovered a new ( free, open source ) tool for graphs and networks. What Photoshop is for pictures, Gephi ( supposedly ) is for graphs. Software is definitely a requirement for Graph Theory. I haven't looked at Gephi in-depth yet, I might. It is exciting though. There is so much going on. When you turn away from a certain area for just a brief period things might have changed quite a lot when you get back. I 'worked with' ( read: used to study graph theory ) Mathematica with the Combinatorica package which is really a gem by itself.

Link: Gephi

Solving the Königsberg Bridge Problem with Mathematica

One of the most famous problems in the history of mathematics is the Königsberg Bridge Problem because it clearly marks the beginning of Graph Theory.

"... In addition to that branch of geometry which is concerned with magnitudes, and which has always received the greatest attention, there is another branch, previously almost unknown, which Leibniz first mentioned, calling it the geometry of position. ..."
Soluti problematis ad geometriam situs pertinentis, Euler 1736

The Königsberg bridge problem asks if the seven bridges of the city of Konigsberg (*) over the river Preger can all be traversed in a single trip without doubling back, with the additional requirement that the trip ends in the same place it began. Euler proved in 1736 that there is no such traversal. (*) Königsberg = Kaliningrad in the Russian exclave between Poland and Lithuania.

Using Graph Theory the problem is equivalent to asking if the multigraph on four nodes and seven edges (see figure) has an Eulerian cycle. Using Mathematica we would model each bridge as an edge and the parts of the city as a vertex. There are different ways to create a graph in Mathematica, but this problem suits the method of creating a graph from an adjacency matrix. Since EulerianQ returns false we know that Mathematica confirms Euler's original answer.