Graph theory

In mathematics and computer science, graph theory is the study of graphs, mathematical structures used to model pairwise relations between objects from a certain collection. A "graph" in this context refers to a collection of vertices and a collection of edges that connect pairs of vertices. A graph may be undirected, meaning that there is no distinction between the two vertices associated with each edge, or its edges may be directed from one vertex to another


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The Seven Bridges of Königsberg is a famous solved mathematics problem inspired by an actual place and situation. The city of Königsberg, Prussia (now Kaliningrad, Russia) is set on the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges. The question is whether it is possible to walk with a route that crosses each bridge exactly once.
In 1736, Leonhard Euler proved that it was not possible. In proving the result, Euler formulated the problem in terms of graph theory, by abstracting the case of Königsberg

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An Eulerian path, Eulerian trail or Euler walk in an undirected graph is a path that uses each edge exactly once. If such a path exists, the graph is called traversable.
An Eulerian cycle, Eulerian circuit or Euler tour in an undirected graph is a cycle that uses each edge exactly once. If such a cycle exists, the graph is called Eulerian or unicursal.

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In algebraic topology, the Euler characteristic is a topological invariant, a number that describes one aspect of a topological space's shape or structure. It is commonly denoted by χ (Greek letter chi).

The Euler characteristic χ was classically defined for polyhedra, according to the formula

where V, E, and F are respectively the numbers of vertices (corners), edges and faces in the given polyhedron.