On the dot product of graphs over monogenic semigroups.

Now define S a cartesian product of finite times with S M n which is a finite semigroup having elements { 0 , x , x 2 , … , x n } of order n . Γ( S ) is an undirected graph whose vertices are the nonzero elements of S . It is a new graph type which is the dot product. k be finite positive integer for 0 ≤ { i t } t = 1 k , { j t } t = 1 k ≤ n , any two distinct vertices of S ( x i 1 , x i 2 , … , x i k ) and ( x j 1 , x j 2 , … , x j k ) are adjacent if and only ( x i 1 , x i 2 , … , x i k ) · ( x j 1 , x j 2 , … , x j k ) = 0 S M n (under the dot product) and it is assumed x i t = 0 S M n if i t = 0 . In this study, the value of diameter, girth, maximum and minimum degrees, domination number, clique and chromatic numbers and in parallel with perfectness of Γ( S ) are elucidated. [ABSTRACT FROM AUTHOR]