Complete Graph & Connected Component
Most graph libraries will find the connected components of a graph G as a list of connected subgraphs [H1, H2, H3, ...] and most of these libraries will have the functionality to obtain the number of nodes and number of edges in a graph. Assuming H is a simple graph (no loops, no multiple edges) with m edges and n vertices, one can say it is complete if

m == n*(n-1)/2


So, given the output of connected components, you should do:

for each graph H in [H1, H2, ...]
  int n = H.getNumVertices()
  int m = H.getNumEdges()
  if (n*(n-1) == 2*m)
    state that H is complete
  else
    state that H is not complete