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Abstract  

We give constructive necessary and sufficient conditions for the validity of the generalized rank reduction formula of Ouellette [12]:

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$r(A - AN(M^T AN)^ - M^T A) = r(A) - r(AM(M^T AN)^ - M^T ).$$ \end{document}
We also analyze the sufficient conditions of Takane and Yanai [16] showing that some of them are also necessary.

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Abstract  

The basis number of a graph G is defined to be the least positive integer d such that G has a d-fold basis for the cycle space of G. We investigate the basis number of the cartesian product of stars and wheels with ladders, circular ladders and Möbius ladders.

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