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

In (Das et al. in J. Inequal. Appl. 2013:44, 2013), a new graph Γ (S_M) on monogenic semigroups S_M (with zero) having elements {0, x, x², x³, ⋯ , xⁿ} was 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 x^j are adjacent if x ⁱx^j = 0 in S_M. As a continuing study, in an unpublished work, some well-known indices (first Zagreb index, second Zagreb index, Randić index, geometric-arithmetic index, atom-bond connectivity index, Wiener index,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 (S_M) were investigated by the same authors of this paper. In the light of the above references, our main aim in this paper is to extend these studies to the lexicographic product over Γ (S_M). In detail, we investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the lexicographic product of any two (not necessarily different) graphs Γ (S¹_M) and Γ (S²_M).