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Let us consider groups G1=Zk∗(Zm∗Zn)Open image in new window, G2=Zk×(Zm∗Zn)Open image in new window, G3=Zk∗(Zm×Zn)Open image in new window, G4=(Zk∗Zl)∗(Zm∗Zn)Open image in new window and G5=(Zk∗Zl)×(Zm∗Zn)Open image in new window, where k,l,m,n≥2Open image in new window. In this paper, by defining a new graph Γ(Gi)Open image in new window based on the Gröbner-Shirshov bases over groups GiOpen image in new window, where 1≤i≤5Open image in new window, we calculate the diameter, maximum and minimum degrees, girth, chromatic number, clique number, domination number, degree sequence and irregularity index of Γ(Gi)Open image in new window. Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in such fields as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics. In addition, the Gröbner-Shirshov basis and the presentations of algebraic structures contain a mixture of algebra, topology and geometry within the purposes of this journal.

The graph based on Gröbner-Shirshov bases of groups