Some properties on the tensor product of graphs obtained by monogenic semigroups

In Das et al. (2013) [8], a new graph Γ(^SM) on monogenic semigroups ^SM (with zero) having elements {0,x,^x2, ^x3,⋯,^xn} has been recently defined. The vertices are the non-zero elements x,^x2,^x3,⋯,^xn and, for 1≤i,j≤n, any two distinct vertices ^xi and ^xj are adjacent if ^xixj=0 in ^SM. As a continuing study, in Akgunes et al. (2014) [3], it has been investigated some well known indices (first Zagreb index, second Zagreb index, Randić index, geometric-arithmetic index, atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, the degree distance) over Γ(^SM). In the light of above references, our main aim in this paper is to extend these studies over Γ(^SM) to the tensor product. In detail, we will investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the tensor product of any two (not necessarily different) graphs Γ(SM1) and Γ(SM2).