Fix $n$ and let $B, C$ be two $n \times n$ 0-1 matrices of full rank such that $\sum\_{i,j} b\_{i,j}^2 = \sum\_{i,j} c\_{i,j}^2$, in other words they have the same number of $0$ entries and the same number of $1$ entries. I want to find $\alpha\_n = \max ||B||/||C||$ subject to this constraint as a function of $n$. 



Maximizing/Minimizing the Operator norms of 0-1 matrices subject to a constraint
Take $B$ to have $1$s on the main diagaonal and the first superdiagonal, then $\|B\|\leq 2$. If we take $C$ to have $1$s on the diagonal and filling the last column, then $\|C\|\geq \sqrt{n}$. (These both have full rank and $2n-1$ $1$'s.) This shows that $\alpha\_n\geq \frac{\sqrt{n}}{2}$, though from the examples in Robert Israel's answer this bound is not optimal.


EDIT: For example when $n=5$:


\begin{equation\*}
B= \begin{pmatrix} 1 & 1 & 0 & 0 & 0\\ 0 & 1 & 1 & 0 & 0 \\ 0 & 0 & 1 & 1 & 0\\ 0 & 0 & 0 & 1& 1 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix} , \quad C= \begin{pmatrix} 1 & 0 & 0 & 0 & 1 \\ 0 & 1 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 0 & 1\end{pmatrix}
\end{equation\*}


