Zagreb Co-Indices on Some Graph Operations of Tadpole Graph

Journal of Mathematics

Topological indices are graph-theoretic parameters which are widely used in the subject of chemistry and computer science to predict the various chemical and structural properties of the graphs respectively. Let G be a graph; then, by performing subdivision-related operations S , Q , R , and T on G , the four new graphs S G (subdivision graph), Q G (edge-semitotal), R G (vertex-semitotal), and T G (total graph) are obtained, respectively. Furthermore, for two simple connected graphs G and H , we define F -sum graphs (denoted by G + F H ) which are obtained by Cartesian product of F G and H , where F ∈ S , R , Q , T . In this study, we determine first general Zagreb co-index of graphs under operations in the form of Zagreb indices and co-indices of their basic graphs.

Discrete Applied Mathematics, 2010

Recently introduced Zagreb coindices are a generalization of classical Zagreb indices of chemical graph theory. We explore here their basic mathematical properties and present explicit formulae for these new graph invariants under several graph operations.

The aim of this paper is to show Zagreb co-indices, an important invariant of a graph changes with several graph operators on the subdivision graphs of some connected graphs. Also, derived expressions for the relationship connecting Zagreb co-indices on three graph operators.

2019

Let G be a simple graph. The subdivision graph and the double graph are the graphs obtained from a given graph G which have several properties related to the properties of G. In this paper, the first and second Zagreb and multiplicative Zagreb indices of double graphs, subdivision graphs, double graphs of the subdivision graphs and subdivision graphs of the double graphs of G are obtained. In particular, these numbers are calculated for the frequently used null, path, cycle, star, complete, complete bipartite or tadpole graph.

Opuscula Mathematica, 2016

In this note, we obtain the expressions for multiplicative Zagreb indices and coindices of derived graphs such as a line graph, subdivision graph, vertex-semitotal graph, edge-semitotal graph, total graph and paraline graph.

Applied Mathematics and Nonlinear Sciences

Many chemically important graphs can be obtained from simpler graphs by applying different graph operations. Graph operations such as union, sum, Cartesian product, composition and tensor product of graphs are among the important ones. In this paper, we introduce a new invariant which is named as the first general Zagreb coindex and defined as M ¯ 1 α ( G ) = ∑ uv ∈ E ( G ¯ ) [ d G ( u ) α + d G ( v ) α ] \overline{M}^\alpha_1(G)=\Sigma_{uv\in E(\overline{G})}[d_G(u)^\alpha+d_G(v)^\alpha] , where α ∈ ℝ, α ≠ 0. Here, we study the basic properties of the newly introduced invariant and its behavior under some graph operations such as union, sum, Cartesian product, composition and tensor product of graphs and hence apply the results to find the first general Zagreb coindex of different important nano-structures and molecular graphs.

The reformulated first Zagreb index is the edge version of first Zagreb index of chemical graph theory. The aim of this paper is to obtain an expression for the reformulated first Zagreb index of the some class of graphs such as Tadpole graph, Wheel graph, Ladder graph. Further we also obtain the reformulated first Zagreb index of the line graph, subdivision graph and line graph of subdivision graph for class of graphs.

Journal of Inequalities and Applications, 2013

Recently, 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: 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

Opuscula Mathematica, 2013

For a (molecular) graph G with vertex set V (G) and edge set E(G), the first and second Zagreb indices of G are defined as M1(G) = v∈V (G) dG(v) 2 and M2(G) = uv∈E(G) dG(u)dG(v), respectively, where dG(v) is the degree of vertex v in G. The alternative expression of M1(G) is uv∈E(G) (dG(u) + dG(v)). Recently Ashrafi, Došlić and Hamzeh introduced two related graphical invariants M 1(G) = uv / ∈E(G) (dG(u)+dG(v)) and M 2(G) = uv / ∈E(G) dG(u)dG(v) named as first Zagreb coindex and second Zagreb coindex, respectively. Here we define two new graphical invariants Π1(G) = uv / ∈E(G) (dG(u) + dG(v)) and Π2(G) = uv / ∈E(G) dG(u)dG(v) as the respective multiplicative versions of M i for i = 1, 2. In this paper, we have reported some properties, especially upper and lower bounds, for these two graph invariants of connected (molecular) graphs. Moreover, some corresponding extremal graphs have been characterized with respect to these two indices.

2012

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.