A graph $G$ has an even number of perfect matchings if and only if $\mathrm{\exists}S\subseteq V(G);(S\ne \varphi )$ such that all vertices in $V(G)$ are adjacent to an even number of vertices in $S$.

At the moment, all I can see is just that finding determinant of $A(G)$ over ${F}_{2}$ reveals the parity of perfect matchings.

At the moment, all I can see is just that finding determinant of $A(G)$ over ${F}_{2}$ reveals the parity of perfect matchings.