Given symmetric $A$ denote $A_{skew}$ to be matrix where $A_{skew,ij}=A_{ij}=-A_{skew,ji}$.
Supposing if we have two adjacency matrices $A$ and $B$ related by $A=PBP'$ for a certain collection of permutations $P$ could we say anything about $Q$ in $A_{skew}=QB_{skew}Q'$?