Ahmet Sinan Cevik

The multiplicative Zagreb indices of graph operations

AbstractRecently, Todeschini et al. (Novel Molecular Structure Descriptors - Theory and Applications I, pp. 73-100, 2010), Todeschini and Consonni (MATCH Commun. Math. Comput. Chem. 64:359-372, 2010) have proposed the multiplicative variants of ordinary Zagreb indices, which are defined as follows: ∏1=∏1(G)=∏v∈V(G)dG(v)2,∏2=∏2(G)=∏uv∈E(G)dG(u)dG(v). These two graph invariants are called multiplicative Zagreb indices by Gutman (Bull. Soc. Math. Banja Luka 18:17-23, 2011). In this paper the upper bounds on the multiplicative Zagreb indices of the join, Cartesian product, corona product, composition and disjunction of graphs are derived and the indices are evaluated for some well-known graphs.MSC:05C05, 05C90, 05C07.

On the Harary index of graph operations

The Harary index is defined as the sum of reciprocals of distances between all pairs of vertices of a connected graph. In this paper, expressions for the Harary indices of the join, corona product, Cartesian product, composition and disjunction of graphs are derived and the indices for some well-known graphs are evaluated. In derivations some terms appear which are similar to the Harary index and we name them the second and third Harary index.MSC:05C05, 05C07, 05C90.

On the Kirchhoff matrix, a new Kirchhoff index and the Kirchhoff energy

AbstractAbstractThe main purpose of this paper is to define and investigate the Kirchhoff matrix, a new Kirchhoff index, the Kirchhoff energy and the Kirchhoff Estrada index of a graph. In addition, we establish upper and lower bounds for these new indexes and energy. In the final section, we point out a new possible application area for graphs by considering this new Kirchhoff matrix. Since graph theoretical studies (including graph parameters) consist of some fixed point techniques, they have been applied in the fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory, and physics. MSC: 05C12, 05C50, 05C90.

A new approach to connect algebra with analysis: relationships and applications between presentations and generating functions

For a minimal group (or monoid) presentation P, let us suppose that P satisfies the algebraic property of either being efficient or inefficient. Then one can investigate whether some generating functions can be applied to it and study what kind of new properties can be obtained by considering special generating functions. To establish that, we will use the presentations of infinite group and monoid examples, namely the split extensions Zn⋊Z and Z2⋊Z, respectively. This study will give an opportunity to make a new classification of infinite groups and monoids by using generating functions.MSC:11B68, 11S40, 12D10, 20M05, 20M50, 26C05, 26C10.

New bounds for Randic and GA indices

The main goal of this paper is to present some new lower and upper bounds for the Randic and GA indices in terms of Zagreb and modified Zagreb indices.MSC: 05C05, 05C20, 05C90.

The number of spanning trees of a graph

AbstractLet G be a simple connected graph of order n, m edges, maximum degree Δ1 and minimum degree δ. Li et al. (Appl. Math. Lett. 23:286-290, 2010) gave an upper bound on number of spanning trees of a graph in terms of n, m, Δ1 and δ: t(G)≤δ(2m−Δ1−δ−1n−3)n−3. The equality holds if and only if G≅K1,n−1, G≅Kn, G≅K1∨(K1∪Kn−2) or G≅Kn−e, where e is any edge of Kn. Unfortunately, this upper bound is erroneous. In particular, we show that this upper bound is not true for complete graph Kn.In this paper we obtain some upper bounds on the number of spanning trees of graph G in terms of its structural parameters such as the number of vertices (n), the number of edges (m), maximum degree (Δ1), second maximum degree (Δ2), minimum degree (δ), independence number (α), clique number (ω). Moreover, we give the Nordhaus-Gaddum-type result for number of spanning trees.MSC:05C50, 15A18.

New distance-based graph invariants and relations among them

The eccentricity of a vertex is the maximum distance from it to another vertex, and the average eccentricity of a graph is the mean eccentricity of a vertex. In this paper we introduce average edge and average vertex-edge mean eccentricities of a graph. Moreover, relations among these eccentricities for trees are provided as well as formulas for line graphs and cartesian product of graphs.

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

In (Das et al. in J. Inequal. Appl. 2013:44, 2013), a new graph Γ(SM) on monogenic semigroups SM (with zero) having elements {0,x,x2,x3,…,xn} was recently defined. The vertices are the non-zero elements x,x2,x3,…,xn and, for 1≤i,j≤n, any two distinct vertices xi and xj are adjacent if xixj=0 in SM. As a continuing study, in an unpublished work, some well-known indices (first Zagreb index, second Zagreb index, Randić 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(SM) were investigated by the same authors of this paper.In the light of the above references, our main aim in this paper is to extend these studies to the lexicographic product over Γ(SM). In detail, we investigate the diameter, radius, girth, maximum and minimum degree, chromatic number, clique number and domination number for the lexicographic product of any two (not necessarily different) graphs Γ(SM1) and Γ(SM2).MSC:05C10, 05C12, 06A07, 15A18, 15A36.

Gröbner-Shirshov bases for extended modular, extended Hecke, and Picard groups

In this paper, Gröbner-Shirshov bases (noncommutative) for extended modular, extended Hecke and Picard groups are considered. A new algorithm for obtaining normal forms of elements and hence solving the word problem in these groups is proposed.

A new graph based on the semi-direct product of some monoids

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(PM). Since graph theoretical studies (including such above graph parameters) consist of some fixed point techniques, they have been applied in fields such as chemistry (in the meaning of atoms, molecules, energy etc.) and engineering (in the meaning of signal processing etc.), game theory and physics.MSC:05C10, 05C12, 05C25, 20E22, 20M05.