Let $\rho(M)$ denote the spectral radius (modulus of the largest eigenvalue) of a square matrix $M$.
I am looking for a characterization or anything else interesting about the set of matrices $A$ having the following properties:
(i) $A$ is (entrywise) nonnegative;
(ii) $A$ is irreducible;
(iii) $\rho(A) \geq 1$;
(iv) if $B$ is obtained from $A$ by deleting row $i$ and column $i$ (for any $i$), then $\rho(B) < 1$.