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Bipartite graphs and inverse sign patterns of strong sign-nonsingular matrices. (English) Zbl 0807.05053
Authors’ abstract: A sign-nonsingular matrix is a square matrix $$X$$ such that each matrix $$Y$$ with the same sign pattern as $$X$$ is nonsingular. If, in addition, the sign pattern of the inverse of $$Y$$ is the same for all $$Y$$, then $$X$$ is a strong sign-nonsingular matrix. A fully indecomposable matrix is a matrix whose associated bipartite graph is connected and (perfect) matching covered. The bipartite graphs of fully indecomposable, strong sign-nonsingular matrices are characterized and a recursive construction is given. This characterization is used to determine the sign patterns of the inverses of fully indecomposable, strong sign-nonsingular matrices and to develop a recognition algorithm for such sign patterns. Those maximal strong sign-nonsingular matrices whose sign patterns are uniquely determined by the sign patterns of their inverses are also characterized in terms of bipartite graphs.

##### MSC:
 05C50 Graphs and linear algebra (matrices, eigenvalues, etc.) 15A09 Theory of matrix inversion and generalized inverses 05C75 Structural characterization of families of graphs
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