Families Of Graphs[Complete,Bipartite and Empty Graphs]

Families of Graphs
Yesterday i began my online tutorials on graph theory for MAT419 and MAT310 respectively,and i was able to look into some "basic concepts in graph theory".
Today i will be giving some examples of families of graphs,there are some
certain types of graphs that play prominent roles in graph theory,a very good example is the complete graph.


Complete Graphs
A complete graph $k_n$ on n-vertices is a simple graph with an edge between every pair of vertices. Examples are the $k_1,k_2,k_3,k_5$,there are so many types of complete graphs.
below is an illustration.

Complete Graphs

Empty Graphs
This is the graph which has vertices and no edges connected to them, meaning there is no connection between all vertices in an empty graph unlike the complete graph which has all vertices connected by edges.

Bipartite Graph
This is a graph in which the vertex set can be partitioned into  2-sets, $v_1$ and $v_2$ such that every edge $u,v\in{E}$ has $u\in{v_1}$ and $v\in{v_2}$.
A very good example of the bipartite graph is the one below,

Bipartite Graph

here we can see that every single edge in the graph is between $v_1$ and $v_2$ which makes the graph bipartite.
Note that in bipartite graph, no edge is in between the same vertex.An edge has to be between two different vertex $v_1$ and $v_2$.

Complete Bipartite graph
This is a graph which has every possible edge between the two sets of vertices. The complete bipartite graph can be denoted to have size $k_{n,m}$ where the first vertex set has size n and second vertex set has size m.
A very good example of the complete bipartite graph is shown below.

Complete Bipartite Graph

A very special case of the bipartite graph is the star graph,which i treated under the topic "basic concepts of graph theory".
There are other families of graphs which i treated earlier on such as the path and cycles,triangular graphs e.t.c