$V$ is set of size $n$ and $\operatorname{Sym}\left(V\right)$ denote the group of permutation of $V$. $E\left(K_V\right)$ denote the set of edges of complete graph with $n$ vertices. For $g\in\operatorname{Sym}\left(V\right)$, $\operatorname{orb}_2\left(g\right)$ denote the number of orbits of $g$ in its action on $E\left(K_V\right)$.
The number of isomorphism classes of graphs with vertex set $V$ is equal to $\dfrac{1}{n!}\sum 2^{\operatorname{orb}_2\left(g\right)}$ where sum runs over $g\in\operatorname{Sym}\left(V\right)$ .
I can not understand the term $\operatorname{orb}_2\left(g\right)$.
In this answer he explain with graph of $4$ vertices can anyone explain me how to find number of orbits for particular permutation in $\operatorname{Sym}\left(V\right)$.
He later explain that each $k$-cycle of $\sigma$ gives rise to $\Big\lfloor\dfrac{k}{2}\Big\rfloor$ orbits of edges between vertices of $k$-cycle.
Can anyone explain with consider particular $\sigma$ how $k$-cycles of $\sigma$ forms $\Big\lfloor\dfrac{k}{2}\Big\rfloor$ orbits of edges between vertices of $k$-cycle. (I can not visualize orbits how does it looks!) All I know about orbit of element (say $x$) is set $\{y*x| y\in G\}$.
