We classify cycles of length $k$ into three subsets:
the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.
Show that, if the cycle $(i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ belongs to $\mathcal{A}_{k}$, then $$\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right) = 0.$$
We classify cycles of length $k$ into three subsets:
\begin{itemize}
\item the set $\mathcal{A}_{k}$, consisting of cycles where at least one edge appears only once;
\item the set $\mathcal{B}_{k}$, consisting of cycles where all edges appear exactly twice;
\item the set $\mathcal{C}_{k}$, consisting of cycles where all edges appear at least twice and there exists at least one that appears at least three times.
\end{itemize}
Show that, if the cycle $(i_{1}, i_{2}, \ldots, i_{k}, i_{1})$ belongs to $\mathcal{A}_{k}$, then
$$\mathbb{E}\left(X_{i_{1}i_{2}} X_{i_{2}i_{3}} \cdots X_{i_{k-1}i_{k}} X_{i_{k}i_{1}}\right) = 0.$$