Linear maps on nonnegative symmetric matrices preserving a given independence number

Abstract

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, and let $J_n$ be the $n\times n$ matrix with all entries equal to one. Given any integers $n,t$ with $2\le t\le n-1$, we prove that a linear map $\phi: S_n^+\rightarrow S_n^+$ satisfies $\phi(J_n-I_n)=J_n-I_n$ and $$\alpha(\phi(X))=t~\iff \alpha(X)=t {~~~~\rm for~ all~}X\in S_n^+$$ if and only if there is a permutation matrix $P$ such that $$\phi(X)=P^TXP{~~~~\rm for~ all~}X\in S_n^+.$