$G^{+}$ Method in Action: New Classes of Nonnegative Matrices with Results

The $G^{+}$ method is a new method, a powerful one, for the study of (homogeneous and nonhomogeneous) products of nonnegative matrices – for problems on the products of nonnegative matrices. To study such products, new classes of matrices are introduced: that of the sum-positive matrices, that of the $\left[\mathrm{\Delta }\right]$-positive matrices on partitions (of the column index sets), that of the $g_k^{+}$-matrices… On the other hand, the $g_k^{+}$-matrices lead to necessary and sufficient conditions for the $k$-connected graphs. Using the $G^{+}$ method, we prove old and new results (Wielandt Theorem and a generalization of it, etc.) on the products of nonnegative matrices – mainly, sum-positive, $\left[\mathrm{\Delta }\right]$-positive on partitions, irreducible, primitive, reducible, fully indecomposable, scrambling, or Sarymsakov matrices, in some cases the matrices being, moreover, $g_k^{+}$-matrices (not only irreducible).