A. Sinan Çevik

Gröbner-Shirshov bases of some monoids

The main goal of this paper is to define Grobner-Shirshov bases for some monoids. Therefore, after giving some preliminary material, we first give Grobner-Shirshov bases for graphs and Schutzenberger products of monoids in separate sections. In the final section, we further present a Grobner-Shirshov basis for a Rees matrix semigroup.

The efficiency of standard wreath product

Let £ be the set of all finite groups that have efficient presentations. In this paper we give sufficient conditions for the standard wreath product of two ^-groups to be a £-group.

Analysis approach to finite monoids

In a previous paper by the authors, a new approach between algebra and analysis has been recently developed. In detail, it has been generally described how one can express some algebraic properties in terms of special generating functions. To continue the study of this approach, in here, we state and prove that the presentation which has the minimal number of generators of the split extension of two finite monogenic monoids has different sets of generating functions (such that the number of these functions is equal to the number of generators) that represent the exponent sums of the generating pictures of this presentation. This study can be thought of as a mixture of pure analysis, topology and geometry within the purposes of this journal.AMS Subject Classification:11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.

Gröbner-Shirshov Bases of the Generalized Bruck-Reilly ∗-Extension

In this paper we first define a presentation for the generalized Bruck-Reilly ∗-extension of a monoid and then we work on its Grobner-Shirshov bases.

On a graph of monogenic semigroups

Let us consider the finite monogenic semigroup SM with zero having elements {x,x2,x3,…,xn}. There exists an undirected graph Γ(SM) associated with SM 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(SM) 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(SM1) and Γ(SM2), we present the spectral properties to the Cartesian product Γ(SM1)□Γ(SM2).MSC:05C10, 05C12, 06A07, 15A18, 15A36.

Knit Products of Some Groups and Their Applications

Let G be a group with subgroups A and K (not necessarily normal) such that G = AK and A ∩ K = {1}. Then G is isomorphic to the knit product, that is, the “two-sided semidirect product” of K by A. We note that knit products coincide with Zappa-Szep products (see [18]). In this paper, as an application of [2, Lemma 3.16], we first define a presentation for the knit product G where A and K are finite cyclic subgroups. Then we give an example of this presentation by considering the (extended) Hecke groups. After that, by defining the Schur multiplier of G, we present sufficient conditions for the presentation of G to be efficient. In the final part of this paper, by examining the knit product of a free group of rank n by an infinite cyclic group, we give necessary and sufficient conditions for this special knit product to be subgroup separable. 2000 Mathematics Subject Classification: 20E22, 20F05, 20F55, 20F32.

THE p-COCKCROFT PROPERTY OF CENTRAL EXTENSIONS OF GROUPS

A presentation for an arbitrary group extension is well known. A generalization of the work by Conway et al. (Group Tensor1972, 25, 405–418) on central extensions has been given by Baik et al. (J. Group Theor.). As an application of this we discuss necessary and sufficient conditions for the presentation of the central extension to be p-Cockcroft, where p is a prime or 0. Finally, we present some examples of this result.

On the first Zagreb index and multiplicative Zagreb coindices of graphs

Abstract For a (molecular) graph G with vertex set V (G) and edge set E(G), the first Zagreb index of G is defined as , where dG(vi) is the degree of vertex vi in G. Recently Xu et al. introduced two graphical invariants and named as first multiplicative Zagreb coindex and second multiplicative Zagreb coindex, respectively. The Narumi-Katayama index of a graph G, denoted by NK(G), is equal to the product of the degrees of the vertices of G, that is, NK(G) = . The irregularity index t(G) of G is defined as the number of distinct terms in the degree sequence of G. In this paper, we give some lower and upper bounds on the first Zagreb index M1(G) of graphs and trees in terms of number of vertices, irregularity index, maxi- mum degree, and characterize the extremal graphs. Moreover, we obtain some lower and upper bounds on the (first and second) multiplicative Zagreb coindices of graphs and characterize the extremal graphs. Finally, we present some relations between first Zagreb index and Narumi-Katayama index, and (first and second) multiplicative Zagreb index and coindices of graphs.

A generalization for the clique and independence numbers

In this paper, lower and upper bounds for the clique and independence numbers are established in terms of the eigenvalues of the signless Laplacian matrix of a given graph G.

Some properties on the tensor product of graphs obtained by monogenic semigroups

Abstract In Das et al. (2013) [8], a new graph Γ ( S M ) on monogenic semigroups S M (with zero) having elements { 0 , x , x 2 , x 3 , … , x n } has been recently defined. The vertices are the non-zero elements x , x 2 , x 3 , … , x n and, for 1 ⩽ i , j ⩽ n , any two distinct vertices x i and x j are adjacent if x i x j = 0 in S M . As a continuing study, in Akgunes et al. (2014) [3], it has been investigated some well known indices (first Zagreb index, second Zagreb index, Randic index, geometric–arithmetic index, atom-bond connectivity index, Wiener index, Harary index, first and second Zagreb eccentricity indices, eccentric connectivity index, the degree distance) over Γ ( S M ) . In the light of above references, our main aim in this paper is to extend these studies over Γ ( S M ) to the tensor product. In detail, we will investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the tensor product of any two (not necessarily different) graphs Γ ( S M 1 ) and Γ ( S M 2 ) .