= Short-cycle expectation in a uniform regular graph
{title2=$\mathbb EC_\ell\longrightarrow(r-1)^\ell/(2\ell)$}
For fixed $r,\ell\geq3$ and $rn$ even, a prescribed <graph cycle> in the <configuration model> has two ordered distinct half-edges at each of its vertices. Therefore its expected count is $(n)_\ell[r(r-1)]^\ell/[2\ell\prod_{j=0}^{\ell-1}(rn-2j-1)]$. Conditioning on the chosen cycle leaves degrees $r-2$ at its vertices and $r$ elsewhere; forbid further edges along that cycle. The <bounded-degree pairing avoidance estimate> changes by $o(1)$, so the simplicity probability conditional on the cycle divided by the unconditional probability tends to one. The displayed expectation therefore holds for the uniform <simple graph>, not just the unconditioned pairing model.
Back to article page
