Applications of the Gordan - Stiemke Theorem in Combinatorial Matrix Theory

Abstrad. By u~ of the Gordan-Stiemke Theorem of the alternative we demonstrate the similarity of four theorems in combinatorial matrix theory. Each theorem contains five equivalent conditions, one of which is the existence in a given pattern of a line-sum-symmetric or constant-line-sum matrix which is semi-positive or strictly positive for the pattern. A generalization of the Gordan-Stiemke Theorem is stated in terms of complementary faces of the positive orthant and combinatorial applications are given. Many of our results are classical, but some may be new.