Types of Graphs:
1. Null Graph: A null graph is defined as a graph which consists only the isolated vertices.
Example: The graph shown in fig is a null graph, and the vertices are isolated vertices.
2. Undirected Graphs: An Undirected graph G consists of a set of vertices, V and a set of edge E. The edge set contains the unordered pair of vertices. If (u, v)∈E then we say u and v are connected by an edge where u and v are vertices in the set V.
Example: Let V = {1, 2, 3, 4} and E = {(1, 2), (1, 4), (3, 4), (2, 3)}.Draw the graph.
Solution: The graph can be drawn in several ways.
Two of which are as follows:
3. Multigraph: If in a graph multiple edges between the same set of vertices are allowed, it is known as Multigraph. In other words, it is a graph having at least one loop or multiple edges.
4. Directed Graphs: A directed graph or digraph G is defined as an unordered pair (V, E), where V is the set of points called vertices and E is the set of edges. Each edge in the graph G is assigned a direction and is identified with an ordered pair (u, v), where u is the initial vertex, and v is the end vertex.
Example: Consider the graph G = (V, E) as shown in fig. Determine the vertex set and edge set of graph G.
Solution: The vertex and edge set of graph G =(V, E) is as follow
        G={{1,2,3},{(1,2),(2,1),(2,2),(2,3),(1,3)}}.
5. Undirected Complete Graph: An undirected complete graph G=(V,E) of n vertices is a graph in which each vertex is connected to every other vertex i.e., and edge exist between every pair of distinct vertices. It is denoted by Kn.A complete graph with n vertices will have 
edges.
Example: Draw Undirected Complete Graphs k4and k6.
Solution: The undirected complete graph of k4 is shown in fig1 and that of k6is shown in fig2.
6. Connected and Disconnected Graph:
Connected Graph: A graph is called connected if there is a path from any vertex u to v or vice-versa.
Disconnected Graph: A graph is called disconnected if there is no path between any two of its vertices.
Example: Consider the graph shown in fig. Determine whether the graphs are
(a)Disconnected Graph
(b)Connected Graph.
Also, write their connected components.
Solution:
(i) The graph is shown in fig is a Disconnected Graph, and its connected components are
      {V1,V2,V3,V4},{V5,V6,V7,V8} and {V9,V10}.
(ii) The graph shown in fig is a Disconnected Graph and its connected components are
      {V1,V2},{V3,V4},{V5,V6},{V7,V8},{V9,V10}and {V11,V12}.
(iii) The graph shown in fig is a connected graph.
7. Connected Component: A subgraph of graph G is called the connected component of G, if it is not contained in any bigger subgraph of G, which is connected. It is defined by listing its vertices.
Example: Consider the graph shown in fig. Determine its connected components.
Solution: The connected components of this graph is {a, b, c}, {d, e, f}, {g, h ,i} and {j}.
8. Directed Complete Graph: A directed complete graph G = (V, E) on n vertices is a graph in which each vertex is connected to every other vertex by an arrow. It is denoted by Kn.
Example: Draw directed complete graphs K3 and K5.
Solution: Place the number of vertices at the appropriate place and then draw an arrow from each vertex to every other vertex as shown in fig:
9. Complementary Graph: The complement of a graph G is defined to be a graph which has the same number of vertices as in graph G and has two vertices connected if and only they are not related in the graph.
Example: Consider the graph G shown in fig. Find the complement of this graph.
Solution: The complement of the above graph is shown in Fig:
10. Labeled Graphs: A graph G=(V, E) is called a labeled graph if its edges are labeled with some name or data. So, we can write these labels in place of an ordered pair in its edges set.
Example: The graph shown in fig is labeled graphs.
      G= {{a, b, c, d}, {e1,e2,e3,e4}}
11.Weighted Graphs: A graph G=(V, E) is called a weighted graph if each edge of graph G is assigned a positive number w called the weight of the edge e.
Example: The graph shown in fig is a Weighted Graph.
