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 $[0,1]$.
Help with the bound (lower and upper) for the largest eigenvalue of such random matrices.
Gershgorin circle could help with the upper bound. For example, if we assume all those $1$s are in the same row, then we should be able to find the probabilistic bound for this case with Irwin–Hall distribution; but I currently have trouble dealing with the "randomly scattered" $1$s.
I am not familiar with the random matrix theory. I am not sure if there is anything from it can help this.