We study n*n random symmetric matrices
whose entries above the diagonal are iid random variables each of which takes 1 with probability p and 0 with probability 1-p, 
for a given density parameter p=alpha/n for sufficiently large alpha.
For a given such matrix A,
we consider a matrix A'
that is obtained by
removing some rows and corresponding columns with too many value 1 entries.
Then for this A',
we show that
the largest eigenvalue is asymptotically close to alpha+1 and its eigenvector is almost parallel to all one vector (1,...,1).