Matrix cut-norms and their relations to graphs

Graph Theory Seminar
Thursday, April 29, 2010 - 12:05pm
1 hour (actually 50 minutes)
Skiles 255
University of Memphis
In 1997 Kannan and Frieze defined the \emph{cut-norm} $\left\Vert A\right\Vert_{\square}$ of a $p\times q$ matrix $A=\left[ a_{ij}\right] $ as%\[\left\Vert A\right\Vert _{\square}=\frac{1}{pq}\max\left\{ \left\vert\sum_{i\in X}\sum_{j\in Y}a_{ij}\right\vert :X\subset\left[ p\right],Y\subset\left[ q\right] ,\text{ }X,Y\neq\varnothing\right\} .\]More recently, Lov\'{a}sz and his collaborators used the norm $\left\VertA\right\Vert _{\square}$ to define a useful measure of similarity between anytwo graphs, which they called \emph{cut-distance. }It turns out that the cut-distance can be extended to arbitrary complexmatrices, even non-square ones. This talk will introduce the basics of thecut-norm and \ cut-distance for arbitrary matrices, and present relationsbetween these functions and some fundamental matricial norms, like theoperator norm. In particular, these relations give a solution to a problem of Lov\'{a}sz.Similar questions are discussed about the related norm\[\left\Vert A\right\Vert _{\boxdot}=\max\left\{ \frac{1}{\sqrt{\left\vertX\right\vert \left\vert Y\right\vert }}\left\vert \sum_{i\in X}\sum_{j\inY}a_{ij}\right\vert :X\subset\left[ p\right] ,Y\subset\left[ q\right],\text{ }X,Y\neq\varnothing\right\} .\]which plays a central role in the \textquotedblleft expander mixinglemma\textquotedblright.