A Further Note on the Graph of Monogenic Semigroups

Abstract

In [15], it has been recently defined a new graph $\Gamma ({% \mathcal{S}}_{M})$ on monogenic semigroups ${\mathcal{S}}_{M}$ (with zero) having elements $\{0,x,x^{2},x^{3},\cdots ,x^{n}\}$. The vertices are the non-zero elements $x,x^{2},x^{3},\cdots ,x^{n}$ and, for $1\leq i,j\leq n$, 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 [3] and [15], 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 $\Gamma ({\mathcal{S}}_{M})$.