The graph based on Gröbner-Shirshov bases of groups

Eylem G Karpuz,Firat Ates,A Sinan Çevik,I Naci Cangul

doi:10.1186/1687-1812-2013-71

Abstract

Let us consider groups G 1 = Z k ∗( Z m ∗ Z n ), G 2 = Z k ×( Z m ∗ Z n ), G 3 = Z k ∗( Z m × Z n ), G 4 =( Z k ∗ Z l )∗( Z m ∗ Z n ) and G 5 =( Z k ∗ Z l )×( Z m ∗ Z n ), where k,l,m,n≥2. In this paper, by defining a new graph Γ( G i ) based on the Gröbner-Shirshov bases over groups G i , where 1≤i≤5, we calculate the diameter, maximum and minimum degrees, girth, chromatic number, clique number, domination number, degree sequence and irregularity index of Γ( G i ). 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.MSC:05C25, 13P10, 20M05, 20E06, 26C10.

Highlights

Introduction and preliminariesIn [, ], the authors have recently developed a new approach between algebra and analysis
In this paper, we investigate the interplay between the group-theoretic property of a group G and the graph-theoretic properties of (G) which is associated with G
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

Summary

Introduction

Introduction and preliminariesIn [ , ], the authors have recently developed a new approach between algebra (in the meaning of groups and monoids) and analysis (in the meaning of generating functions). 2.2 Case 2: the graph (G2), where G2 = Zk × (Zm ∗ Zn) If we consider the graph of the group G , we have a subgraph of Figure (a) with vertices v = am, v = a, v = xka, v = xk , v = x, v = xam, v = xa, v = xkb, v = b, v = bn, v = xbn and v = xb.

Results

Conclusion