The simplest way to do it is to do a depth first traversal (DFT) of the graph.
If the graph has n vertices, this is a O(n) time complexity algorithm. Since you will possibly have to do a DFT starting from each vertex, the total complexity becomes O(n^2).
You have to maintain a stack containing all vertices in the current depth first traversal, with its first element being the root node. If you come across an element which is already in the stack during the DFT, then you have a cycle.
