Notes on the generalized Perron complements involving inverse $N_0$-matrices

In the context of inverse $N_0$-matrices, this study focuses on the closure of generalized Perron complements by utilizing the characteristics of $M$-matrices, nonnegative matrices, and inverse $N_0$-matrices. In particular, we illustrate that the inverse $N_0$-matrix and its Perron complement matrix possess the same spectral radius. Furthermore, we present certain general inequalities concerning generalized Perron complements, Perron complements, and submatrices of inverse $N_0$-matrices. Finally, we provide specific examples to verify our findings.