Summary: In [K. Ch. Das et al., J. Inequal. Appl. 2013, Paper No. 44, 13 p. (2013; Zbl 1283.05067)], it has been recently defined a new graph \(\Gamma(\mathcal{S}_M)\) on monogenic semigroups \(\mathcal{S}_M\) (with zero) having elements \(\{0,x,x^2,x^3,\ldots,x^n\}\).
The vertices are the non-zero elements \(x,x^2,x^3,\ldots,x^n\) and, for \(\leq i,j\leq n\), any two distinct vertices \(x^i\) and \(x^j\) are adjacent if \(x^i x^j=0\) in \(\mathcal{S}_M\). As a continuing study of [N. Akgüneş, K. Ch. Das and A. S. Cevik: “Topological indices on a graph of monogenic semigroups”, in: I. Gutman (ed.), Topics in chemical graph theory. Kragujevac: University of Kragujevac. 3–20 (2014)] and [Das et al., loc. cit.], in this paper it will be investigated some special parameters (such as covering number, accessible number, independence number), first and second multiplicative Zagreb indices, and Narumi-Katayama index. Furthermore, it will be presented Laplacian eigenvalue and Laplacian characteristic polynomial for \(\Gamma(\mathcal{S}_M)\).