A cycle is a simple path of length at least $1$ which begins and ends at the same vertex.
In an undirected graph, a cycle must be of length at least $3$.
Could you explain me why that stands??
[Math] A cycle in an undirected graph
discrete mathematicsgraph theory
Related Solutions
There are 6 possibilities for picking the first vertex of the 4-cycle,
5 possibilities for then picking the second vertex,
4 possibilities for picking the third vertex,
and 3 possibilities for picking the fourth vertex.
Now we have picked every cycle four times rotated (as a cycle is invariant under rotations) and twice reflected (as we traverse the cycle once clockwise and once counter-clockwise).
Hence the answer is $(6\cdot5\cdot4\cdot3)~/~(4\cdot2)=45$.
To tack on to Fuseques' answer that highlights the distinction for simple graphs.
Say we have a walk between vertices $x$ and $y$. If vertex $x$ does not equal to vertex $y$ then it is called a path. However, if $x = y$ then it is called a cycle.
Be a little bit careful with your definitions.
A walk between $2$ vertices $x$ and $y$, commonly referred to as an $xy$-walk, is a sequence of vertices (or vertices and edges, but naming the edges is not necessary) that we can traverse to get from $x$ to $y$. Note that in a walk, edges and vertices can repeat.
A trail is a walk that does not repeat an edge.
A path is a walk that does not repeat vertices (and thus does not repeat edges).
Now some may fuzz this definition a bit to say that that we can have a closed path in order to define cycles. You can see the differences in the way cycles are defined at Wolfram MathWorld and Wikipedia.
A closed walk, i.e. an $xx$-walk is not necessarily a cycle, but a cycle is a closed walk. (See Misha Lavrov's comment).
An example:
A walk (in this case closed) would be $1-2-3-4-5-2-3-4-5-2-1$.
A trail would be $1-2-3-4-5-2$.
A path would be $4-3-2-1-6-9-8$.
Best Answer
In an undirected simple graph, there are no self loops (which are cycles of length 1) or parallel edges (which are cycles of length 2). Thus all cycles must be of length at least 3.