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\section{Nonsingularity of Divisor Tournaments} By {\sl   Rohan Hemasinha Dept. of Math/Stat, Univ. of West Florida, Pensacola, FL 32514, USA rhemasin@uwf.edu Jeffrey L. Stuart Dept. of Mathematics, Pacific Lutheran Univ. Tacoma, WA 98447, USA jeffrey.stuart@plu.edu James R. Weaver (speaker) Dept. of Math/Stat, Univ. of West Florida, Pensacola, FL 32514, USA jweaver@uwf.edu}.

\medskip \noindent   Matrix theoretic properties and examples of divisor tournaments are discussed. Emphasis is placed on results and conjectures about the nonsingularity of the adjacency matrix for a divisor tournament.  
Definition: For an integer $n>2$, the divisor tournament $D(T_{n})$
 ( a directed graph on the vertices $2,3,\ldots,n$) is defined by: $i$
 is adjacent to $j$ if $i$ divides $j$, otherwise $j$ is adjacent to $i$ for
$ 2��i<j��n$. No vertex is adjacent to itself. Definition: 
The adjacency matrix $T_{n}$ of the directed graph $D(T_{n})$ with vertex set $\{2,3,\ldots,n\}$ is the $(n-1)��(n-1)$ matrix $[t_{ij}]$ defined by $t_{ij}=1$ 
and $t_{ji}=0$ if $i�Oj$, $t_{ij}=0 $ and $t_{ji}=1$ if $i \&\# 8740;j$ for $2��i<j��n$. $t_{ii}=0$ for $i  \in \{2,3,\ldots,n\}$.
%","tournament, eigenvalue, eigenvector","15","5","","14:34:53","Thu Mar 20 2008","72.203.164.76"
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