Graph Theory

Graph Theory is a way to model a collection of objects, and the connections between those objects.

$$\text{A graph }G\text{ is a set }V\text{ of vertices together with a set }E\text{ of edges which }\\\text{are unordered pairs of distinct vertices.}$$

The complete graph on $n$ vertices, $K_n$, is defined by taking a set of $n$ vertices and connecting every pair of distinct vertices by an edge.

Complete graphs.

Complete graphs.

The Complete Bipartite Graph $K_{m,n}$ is defined by splitting the vertices into $2$ subsets, one of size $m$ and one of size $n$, and joining all vertices in one subset to all vertices in the other.

Complete bipartite graphs.

Complete bipartite graphs.

The Cycle Graph on $n\geq3$ vertices $\text{C}_n$ is defined by the following: