Breadth First Search time complexity analysis

Question

The time complexity to go over each adjacent edge of a vertex is, say, O(N), where N is number of adjacent edges. So, for V numbers of

Answer 1

You are saying that total complexity should be O(V*N)=O(E). Suppose there is no edge between any pair of vertices i.e. Adj[v] is empty for all vertex v. Will BFS take a constant time in this case? Answer is no. It will take O(V) time(more accurately θ(V)). Even if Adj[v] is empty, running the line where you check Adj[v] will itself take some constant time for each vertex. So running time of BFS is O(V+E) which means O(max(V,E)).