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(SM) on monogenic semigroups SM (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 xj are adjacent if xixj=0 in SM. As a continuing study, in an unpublished work, some well-known indices (first Zagreb index, second Zagreb index, Randić index, geometric-arithmetic index, atom-bond connectivity index, Wiener index,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(SM) were investigated by the same authors of this paper.