  The independence number of a square matrix $A$, denoted by $\alpha(A)$, is
the maximum order of its principal zero submatrices. Let $S_n^{+}$ be the set
of $n\times n$ nonnegative symmetric matrices with zero trace. Denote by $J_n$
the $n\times n$ matrix with all entries equal to one. Given any integer $n$, we
prove that a linear map $\phi: S_n^+\rightarrow S_n^+$ satisfies
$$\alpha(\phi(X))= \alpha(X) {\quad\rm for~ all\quad}X\in S_n^+$$ if and only
if there is a permutation matrix $P$ such that $$\phi(X)=H\circ(P^TXP)\quad {
\rm for~ all\quad}X\in S_n^+,$$ where $H=\phi(J_n-I_n)$ with all off-diagonal
entries positive.
