In this paper, firstly, we define a new graph based on the semi-direct product of a free abelian monoid of rank n by a finite cyclic monoid, and then discuss some graph properties on this new graph, namely diameter, maximum and minimum degrees, girth, degree sequence and irregularity index, domination number, chromatic number, clique number of Γ ( P M ).
In this section, by considering the graph Γ ( P M ) drawn in Figure 1, we mainly deal with some graph properties, namely diameter, maximum and minimum degrees, girth, degree sequence, irregularity index, domination number, chromatic number and clique number of Γ ( P M ).
In this paper, we mainly obtain some certain values for the diameter, girth, maximum and minimum degree, chromatic number, clique number, degree sequence, irregularity index and domination number of the graph Γ ( S M ).
In this section, by considering the graph Γ ( S M ) defined as in the first section, we will mainly deal with the graph properties, namely diameter, girth, maximum and minimum degrees, domination number and finally irregularity index of it.
Manipulation and power domination".
In fact, by the graph-theoretic properties, we will be interested in the diameter, maximum and minimum degrees, girth, chromatic number, clique number, domination number, degree sequence and irregularity index of the corresponding new graph.
By group-theoretic property, while we deal with the Gröbner-Shirshov basis of a given group, by graph-theoretic property, we are interested in the diameter, maximum and minimum degrees, girth, chromatic number, clique number, domination number, degree sequence and irregularity index of the corresponding graph of group.
Of irresistible force, power and domination.
The topological indices of graphs with given domination number or domination variations have attracted much attention of researchers [1, 45 47].
Borovićanin [1] showed the sharp upper bounds on the Zagreb indices of n-vertex trees with domination number γ and characterized the extremal trees.
