Linear maps on nonnegative symmetric matrices preserving the independence number

TL;DR

This paper characterizes linear maps on nonnegative symmetric matrices with zero trace that preserve the independence number, showing they are essentially permutation conjugations combined with a specific entrywise positive matrix.

Contribution

It provides a complete characterization of linear maps preserving the independence number on a class of nonnegative symmetric matrices.

Findings

Such maps are characterized by permutation similarity and entrywise multiplication with a positive matrix.

The result applies to matrices with zero trace and nonnegative symmetric structure.

It extends understanding of structure-preserving linear transformations in matrix theory.

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. 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.