What is a Network in the context of Graph?

Network refers to a graph in which the edges are bound with weights. Here weights can be numbers assigned to a the edges. Hence, a graph of this nature is referred to as weighted graph or a network.

Types of Graphs

There two types of graphs:
i. Undirected Graphs
ii. Directed Graphs

Undirected Graph: A graph that entail edges with ordered pair of vertices, however it does not have direction define. Example of such a graph is the 'Family tree of the Greek gods'

Directed Graph: A graph that entail edges with ordered pair of vertices and has direction indicated with an arrow. Example of such a graph is the 'A finite-state machine of a light switch'

What is a Graph data structure?

Graph is an Abstract Data Type (ADT) which is of a non-linear structure entailing sets of nodes and connectors. Here, the connectors helps to establish relationship between nodes. However, from the graph ADT literature point-of-view the nodes are referred to as Vertices and connectors are referred to as Edges.
Mathematically, a graph G entails vertices V, which are connected by edges E. Hence, a graph is defined as an ordered pair G=(V,E), where V is a finite set and E is a set consisting of two element subsets of V.