Discrete Mathematics

Directed Graph :
A graph G is called the directed graph, the set of vertices are V and the set of edges is E, consists the order pairs of elements of V. In general, we can say that each pair of vertices are connected by a straight line a direction between both the vertices are given.
Let us take the examples, that we have a graph G1, with the set of three vertices i.e. v1, v2 and v3 and the three edges e1, e2 and e3 then G1 = ({v1, v2, v3}, {e1, e2, e3})

Take another example, the graph has 4 vertices and edges i.e. G = ({v1, v2, v3 v4}, {e1, e2, e3, e4})

Undirected Graph :
If in a graph G, the set of vertices are V and the set of edges are E and every edge is associated with unordered pair of vertices V, then a graph G is called as Undirected Graph. In general, we can say that each pair of vertices is connected by a line and direction between two vertices is not there.
Let us take the examples same as above G1 = ({v1, v2, v3}, {e1, e2, e3})

Take another example, the graph has 4 vertices and 4 edge i.e. G = ({v1, v2, v3 v4}, {e1, e2, e3, e4})

Mixed Graph :
This is the combination of both the directed graph and undirected graph i.e. A Graph G is called a mixed graph if some of the edges in a graph are directed and some are undirected.
Let us take the examples same as above G1 = ({v1, v2, v3}, {e1, e2, e3})

Take another example, the graph has 4 vertices and 4 edges G = ({v1, v2, v3, v4}, {e1, e2, e3, e4})

Null Graph :
If in a graph all of the given set of vertex are isolated i.e. no edges, only vertices called the Null Graph.

Important Definition :

What is Isolated Vertex?
Any vertex V in a graph is called the isolated vertex which is not connected or joined by the graph G by the edges (with no edges connected). Then it is called the Isolated Vertex.
Self-Loop Graph :
In a graph G, an edge having the same vertex at both the end of the vertices is called the self-loop graph.
If in a graph the set of vertices are V= (1, 2, 3, 4) and the edges E are {(1, 2), (2, 2), (2, 3), (3, 4)}, the edge (2, 2) has the same starting and end point i.e. at vertex 2, is a self-loop graph.

Simple and Multi Graph :
If in a graph, there is no self-loop and no parallel edges, then the graph is called a simple graph and a multigraph is a graph with multiple edges between the same vertices.

G1 : Simple Graph & G2 is Multiple Graph

Finite and Infinite Graph :
If in a graph G, the set of edges and vertices are finite then the graph is a finite graph else it is an infinite graph.

Connected and Disconnected Graphs :
In a graph, if there is at least one path between every pair of vertices, then a graph is known as a connected else it is Disconnected graph.