Let $A$ be a primitive $N\times N$-matrix with positive entries, that is there is $n>0$ such that $(A^n)_{i,j}>0$ for all $i,j$. For brevity, assume the entries consist only of $0$ and $1$.
The Perron-Frobenius theorem gives, among other things, a distinguished eigenvalue $\lambda_1>1$ that is simple and for all the other eigenvalues $\lambda_j$ it holds that $|\lambda_j|<\lambda_1$.
I would like to ask if there is a characterization of matrices $A$ (or associated strongly connected graphs) for which it holds that $|\lambda_j|\leq 1$, for every $j\neq 1$? Thanks!
