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)


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

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 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.