Aysun Yurttas

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 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.

Effect of Edge Deletion and Addition on Zagreb Indices of Graphs

Edge deletion and addition to a graph is an important combinatorial method in Graph Theory which enables one to calculate some properties of a graph by means of similar graphs. The effect of edge addition on the first and second Zagreb indices was recently investigated by the authors. In this sequel paper, we consider the change in the first and second Zagreb indices of any simple graph G when an arbitrary edge is deleted. Further, we calculate the change in the first Zagreb index when an arbitrary number of edges are deleted. This method can be used to calculate the first and second Zagreb indices of larger graphs in terms of the Zagreb indices of smaller graphs. As some examples, we give some inequalities for the change of Zagreb indices for path, cycle, star, complete, complete bipartite, and tadpole graphs.

New formulae for Zagreb indices

In this paper, we study with some graph descriptors also called topological indices. These descriptors are useful in determination of some properties of chemical structures and preferred to some earlier descriptors as they are more practical. Especially the first and second Zagreb indices together with the first and second multiplicative Zagreb indices are considered and they are calculated in terms of the smallest and largest vertex degrees and vertex number for some well-known classes of graphs.

Deterrmining the minimal polynomial of cos(2π/n) over Q with Maple

The number λq = 2cosπ/q, q ∈ N, q ≥ 3, appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number and in some of these methods, the minimal polynomials of several algebraic numbers are used. Here we obtain the minimal polynomial of one of those numbers, cos(2π/n), over the field of rationals by means of the better known Chebycheff polynomials for odd q and give some of their properties. We calculated this minimal polynomial for n ∈ N by using the Maple language and classifying the numbers n ∈ N into different classes.

Calculation of the minimal polynomial of 2cos(π/n) over Q with Maple

The number λq = 2cosπ/q, q ∈ N, q ≥ 3, appears in the study of Hecke groups which are Fuchsian groups and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number. Here we obtain the minimal polynomial of these numbers over the field of rationals by means of the better known Chebycheff polynomials and the Maple language.

Some formulae for the Zagreb indices of graphs

In this study, we first find formulae for the first and second Zagreb indices and coindices of certain classical graph types including path, cycle, star and complete graphs. Secondly we give similar formulae for the first and second Zagreb coindices.

Some Properties of the Minimal Polynomials of 2cos(π/q) for odd q

The number λq = 2 cos π/q, q∈N, q≥3,, appears in the study of Hecke groups which are Fuchsian groups, and in the study of regular polyhedra. There are many results about the minimal polynomial of this algebraic number. Here we obtain the minimal polynomial of this number by means of the better known Chebycheff polynomials for odd q and give some of their properties.

Upper Bounds for the Level of Normal Subgroups of Hecke Groups

In [4], Greenberg showed that n≤6t3 so that μ = nt≤6t4 for a normal subgroup N of level n and index μ having t parabolic classes in the modular group Γ. Accola, [1], improved these to n≤6t2 always and n≤t2 if Γ/N is not abelian. In this work we generalise these results to Hecke groups. We get results between three parameters of a normal subgroup, i.e. the index μ, the level n and the parabolic class number t. We deal with the case q = 4, and then obtain the generalisation to other q. Two main problems here are the calculation of the number of normal subgroups and the determination of the bounds on the level n for a given t.

Classification of Normal Subgroups of Hecke Group H6 in Terms of Parabolic Class Number

In [3], Greenberg showed that n≤6t3 so that μ = nt≤6t4 for a normal subgroup N of level n and index μ having t parabolic classes in the modular group Γ. Accola, [1], improved these to n≤6t2 always and n≤t2 if Γ/N is not abelian. Newman, [5], obtained another generalisation of these results. Hecke groups are generalisations of the modular group. We particularly deal with one of the most important cases, q = 6.