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(2003 / vol.6 / no.2)
Exponents of $r-$regular primitive matrices
MINGAI JIN, SANG GU LEE, HAN GUK SEOL
Pages. 51-57     



A Boolean matrix $A$ is said to be {f
extit{r$-$regular}} if each vertex in its digraph $G(A)=G$ has
outdegree and indegree exactly $r.$ $A$ is primitive if and only
if there exist minimum integer $k$ such that $A^k>0$. For such a
matrix, its digraph $G$ is strongly connected and given any
ordered pair of vertices $x$ and $y$ there is a directed walk from
$x$ to $y$ of length $k$. Using the graph theory, we determine the
lower bounds and upper bounds of $r-$regular primitive matices in
this paper. We also proved that the exponent of $r-$regular
primitive tournament $T$ is just 3 and the exponent of $r-$regular
primitive symmetric matrix $A$ satisfys ${
m exp}(A)leq 2(n-r).$



1. basic concepts
2. The exponent of $r$-regular primitive matrices
3. Specil $3$-regular primitive matrices