Say $A$ is an $n\times n$ symmetric matrix such that every row (and hence column) has exactly $d<n$ non-zero entries.
Does there exist similarity transformations on $A$ which will maintain these two properties? Does there exist an infinite number of such similarity transformations?
Further suppose the matrix $A$ were such that $1$ was the only non-zero element allowed. Can the above question be answered now? (..surely now an infinite number of such similarity transformations can't exist..)