How do I find the shortest path that covers all nodes in a directed cyclic graph?

Question

I need an example of the shortest path of a directed cyclic graph from one node (it should reach to all nodes of the graph from a node that will be the input).

Pleas

Answer 1

EDIT: Oops, misread the question. Thanks @jfclavette for picking this up. Old answer is at the end.

The problem you're trying to solve is called the Travelling salesman problem. There are many potential solutions, but it's NP-complete so you won't be able to solve for large graphs.

Old answer:

What you're trying to find is called the girth of a graph. It can be solved by setting the distances from a node to itself to infinity and using the Floyd-Warshall algorithm. The length of the shortest cycle from node i is then just the entry in position ii.