Subgraph G'(V',E') of a graph G(V,E) is when V'⊆V and E'⊆E.
Let G has 3 vertices A,B and C.
Subgraphs with one node:
V'={A} and E'=null. There can be 3 such graphs.
Subgraphs with two nodes but no edges:
V'={A,B} and E'=null. There can be 3 such graphs V'={B,C} and {C,A}.
Subgraphs with two nodes and one edge:
V'={A,B} and E'={AB}. 3 such cases.
Subgraphs with 3 nodes and no edge.
V'={A,B,C} and E'=null. 1 case.
Subgraphs with 3 nodes and one edge.
V'={A,B,C} and E'={AB}. Again 3 cases.
Subgraphs with 3 nodes and two edges.
V'={A,B,C} and E'={AB,BC}. 3 cases.
Subgraphs with 3 nodes and 3 edges.
V'={A,B,C} and E'={AB,BC,CA}. 1 such case.
Total subgraphs = 5*3+1+1 =17.