Back Edge : an edge that connects a vertex to a vertex that is it's ancestor.
Start the $BFS$ from vertex $A$ then you take : $AB,AC,AD$, they are tree edges.
Now you take $BC$ it is a cross edge, then $CD$ it is also a cross edge then $DE$ this is a tree edge.
Then you can take $EB$ this is a cross edge.
So, there are no back edges but still a cycle $EBCD$ exists.