On the dot product of graphs over monogenic semigroups

摘要

Now define S a cartesian product of finite times with SMn which is a finite semigroup having elements {0,x,x2,…,xn} 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≤{it}t=1k,{jt}t=1k≤n, any two distinct vertices of S(xi1,xi2,…,xik) and (xj1,xj2,…,xjk) are adjacent if and only (xi1,xi2,…,xik)·(xj1,xj2,…,xjk)=0SMn (under the dot product) and it is assumed xit=0SMn if it=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.