On a graph of monogenic semigroups

Let us consider the finite monogenic semigroup S-M with zero having elements {x, x(2), x(3), ... , x(n)}. There exists an undirected graph Gamma (S-M) associated with S-M whose vertices are the non-zero elements x, x(2), x(3), ... , x(n) 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 Gamma (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 Gamma (S-M(1)) and Gamma (S-M(2)), we present the spectral properties to the Cartesian product Gamma (S-M(1)) square Gamma (S-M(2)).