Can a path contain a cycle .i.e is this diagram a valid path
a-----b----c----d
\ /
\ /
e
According to this definition of path from 'Introduction to Graph Theory by D.B. West' 
So, I ordered this as a,b,c,d,e and call it a path which includes all the edges and vertices in the diagram. path ordering : e,a,b,c,d
Is this wrong, it seems so to me, but don't know why?
Your ordering does not satisfy the given definition, since vertices $d$ and $e$ are not adjacent in the graph, but are adjacent in your list. Indeed, no such ordering exists due to the cycle $a \to b \to e \to a$.