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 $\rho(B) < 1$.
Let's assume the matrix $A$ has all its row sums equal to $\lambda$, the largest eigenvalue. We can rescale the rows and columns of any other nonnegative irreducible matrix by a similarity transformation $A \to D^{-1} A D$ where $D$ is diagonal with positive diagonal entries (namely the entries of the Perron eigenvector of $A$). This makes all row sums equal to $\lambda$, and preserves the eigenvalues of $A$ and of the matrices obtained by removing the $i$'th row and column.
Let $B_k$ be the matrix obtained from $A$ by removing row and column number $k$. The row sums of $B_k$ are $\lambda - a_{ik}$ for $i \ne k$. Thus the largest eigenvalue of $B_k$ is at most $\lambda - \min_{i \ne k} a_{ik}$ and at least $\lambda - \max_{i \ne k} a_{ik}$. So a necessary condition is $$\lambda \ge 1 > \lambda - \min_k \max_{i \ne k} a_{ik}$$ while a sufficient condition is $$\lambda \ge 1 > \lambda - \min_k \min_{i \ne k} a_{ik}$$
It is well-known that the study of irreducible nonnegative matrices is equivalent to studying (row) stochastic matrices (e.g., the paper by Johnson [Row stochastic matrices similar to doubly stochastic matrices. Linear and Multilinear Algebra 10 (1981), no. 2, 113–130; MR0618581]):
