Given a $n\times n$ symmetric random matrix whose diagonals are all fixed as $1$. In addition, there are $k$ $1$s will be randomly scattered in upper triangular (of course, the corresponding places in the lower-triangle will be filled with $1$, and $2k < n^2-n$). All other elements are independent uniform random variables over $[0,1]$.

Is there known bound for the largest eigenvalue of such random matrices?

If there is not, any suggestion of possible method or reference to related materials is very much appreciated.