A perfect matching of a complete graph $K_{2n}$ is a 1-regular subgraph that contains all the vertices. Two perfect matchings intersect if they share an edge. It is known that if $mathcal{F}$ is family of intersecting perfect matchings of $K_{2n}$, then $|mathcal{F}| leq (2(n-1) - 1)!!$ and if equality holds, then $mathcal{F} = mathcal{F}_{ij}$ where $ mathcal{F}_{ij}$ is the family of all perfect matchings of $K_{2n}$ that contain some fixed edge $ij$. We give a short algebraic proof of this result, resolving a question of Godsil and Meagher. Along the way, we show that if a family $mathcal{F}$ is non-Hamiltonian, that is, $m cup m ot cong C_{2n}$ for any $m,m in mathcal{F}$, then $|mathcal{F}| leq (2(n-1) - 1)!!$ and this bound is met with equality if and only if $mathcal{F} = mathcal{F}_{ij}$. Our results make ample use of a somewhat understudied symmetric commutative association scheme arising from the Gelfand pair $(S_{2n},S_2 wr S_n)$. We give an exposition of a few new interesting objects that live in this scheme as they pertain to our results.
تحميل البحث