A Note on the Numbers of Nonzero Entries of Maximal $S^2NS$ Matrices
Received:November 02, 2005  Revised:January 20, 2006
Key Words: sign   maximal   $S^2NS$   matrices   digraphs.  
Fund Project:the National Natural Science Foundation of China (10331020); Tianyuan fund (10526019) and NNSF of Guangdong Province (5300084).
Author NameAffiliation
YOU Li-hua Department of Mathematics, South China Normal University, Guangzhou 510631, China 
SHAO Jia-yu Department of Applied Mathematics, Tongji University, Shanghai 200092, China 
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Abstract:
      A square real matrix $A$ is called an $S^2NS$ matrix, if every matrix with the same sign pattern as $A$ is invertible, and the inverses of all such matrices have the same sign pattern. A matrix $A$ is called a maximal $S^2NS$ matrix, if $A$ is an $S^2NS$ matrix, but each matrix obtained from $A$ by replacing one zero entry by a nonzero entry is not a $S^2NS$ matrix. Let ${\cal S}(n)$ be the set of numbers of nonzero entries of maximal $S^2NS$ matrices with order $n~(\geq 5),$ and $Z_4(n)=\{\frac{1}{2}n(n-1)+4, \cdots, \frac{1}{2}n(n+1)-1\}$. We know that ${\cal S}(n)$ has been described except for the numbers between $2n+1$ and $3n-4$ and the numbers in $Z_4(n)$. We prove $Z_4(n)\cap {\cal S}(n)=\phi$ by graphic method in this paper.
Citation:
DOI:10.3770/j.issn:1000-341X.2007.01.016
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