On a graph of monogenic semigroups

Let us consider the finite monogenic semigroup S_M with zero having elements {x,x²,x³. . . ,xⁿ}. There exists an undirected graph Γ(S_M) associated with S_M whose vertices are the non-zero elements x,x²,x³. . . ,xⁿ and,f or 1 ≤i,j ≤n, any two distinct vertices xi and xj are adjacent if i + j >n. In this paper, the diameter, girth, maximum and minimum degrees, domination number, chromatic number, clique number, degree sequence, irregularity index and also perfectness of Γ(S_M) have been established. In fact, some of the results obtained in this section are sharper and stricter than the results presented in DeMeyer et al. (Semigroup Forum 65: 206-214, 2002). Moreover, the number of triangles for this special graph has been calculated. In the final part of the paper, by considering two (not necessarily different) graphs Γ (S_M ¹) and Γ (S_M²), we present the spectral properties to the Cartesian product Γ (S_M¹ )□ Γ (S _M²).