Consider questions of the form (or the "most probable value of" version of these questions rather than the "largest possible"),

• What is the largest possible spectral radius of a $n \times n$ matrix with entries in $\{0,1,-1 \}$ ?

• What is the largest possible spectral radius of a $n \times n$ matrix with entries in $\{0,1,-1 \}$ and has $d$ non-zero entries ? (or an "at most d" version of this)

• What is the largest possible spectral radius of a $n \times n$ matrix with entries in $\{0,1,-1 \}$ and which has $d$ non-zero entries in every row and/or column? (or an "at most d" version of this)

What methods or techniques we have to answer such things? Any reference would be helpful.

One can restrict the matrices to be symmetric if necessary or to have $0$s along the diagonals if necessary. Or feel free to replace the set $\{0,1,-1\}$ by some other finite set if that makes things better.