Multiplicative Zagreb Indices Research Articles

Topological characterization of dendrimer, benzenoid, and nanocone

Dendrimers are large and complex molecules with very well defined chemical structures. More importantly, dendrimers are highly branched organic macromolecules with successive layers or generations of branch units surrounding a central core. Topological indices are numbers associated with molecular graphs for the purpose of allowing quantitative structure-activity relationships. These topological indices correlate certain physico-chemical properties such as the boiling point, stability, strain energy, and others, of chemical compounds. In this article, we determine hyper-Zagreb index, first multiple Zagreb index, second multiple Zagreb index, and Zagreb polynomials for hetrofunctional dendrimers, triangular benzenoids, and nanocones.

Topological Indices of H-Naphtalenic Nanosheet

Abstract In chemical graph theory, a single numeric number related to a chemical structure is called a topological descriptor or topological index of a graph. In this paper, we compute analytically certain topological indices for H-Naphtalenic nanosheet like Randic index, first Zagreb index, second Zagreb index, geometric arithmetic index, atom bond connectivity index, sum connectivity index and hyper-Zagreb index using edge partition technique. The first multiple Zagreb index and the second multiple Zagreb index of the nanosheet are also discussed in this paper.

Maximizing and Minimizing Multiplicative Zagreb Indices of Graphs Subject to Given Number of Cut Edges

Given a (molecular) graph, the first multiplicative Zagreb index Π 1 is considered to be the product of squares of the degree of its vertices, while the second multiplicative Zagreb index Π 2 is expressed as the product of endvertex degree of each edge over all edges. We consider a set of graphs G n , k having n vertices and k cut edges, and explore the graphs subject to a number of cut edges. In addition, the maximum and minimum multiplicative Zagreb indices of graphs in G n , k are provided. We also provide these graphs with the largest and smallest Π 1 ( G ) and Π 2 ( G ) in G n , k .

Topological Characterizations and Index-Analysis of New Degree-Based Descriptors of Honeycomb Networks

Topological indices and connectivity polynomials are invariants of molecular graphs. These invariants have the tendency of predicting the properties of the molecular structures. The honeycomb network structure is an important type of benzene network. In the present article, new topological characterizations of honeycomb networks are given in the form of degree-based descriptors. In particular, we compute Zagreb and Forgotten polynomials and some topological indices such as the hyper-Zagreb index, first and second multiple Zagreb indices and the Forgotten index, F. We, for the first time, determine some regularity indices such as the Albert index, Bell index and I R M ( G ) index, as well as the F-index of the complement of the honeycomb network and several co-indices related to this network without considering the graph of its complement or even the line graph. These indices are useful for correlating the physio-chemical properties of the honeycomb network. We also give a graph theoretic analysis of some indices against the dimension of this network.

Computing Topological Indices and Polynomials for Line Graphs

A topological index is a number related to the atomic index that allows quantitative structure–action/property/toxicity connections. All the more vital topological indices correspond to certain physico-concoction properties like breaking point, solidness, strain vitality, and so forth, of synthetic mixes. The idea of the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials was set up in the substance diagram hypothesis in light of vertex degrees. These indices are valuable in the investigation of calming exercises of certain compound systems. In this paper, we computed the first and second Zagreb index, the hyper Zagreb index, multiple Zagreb indices and Zagreb polynomials of the line graph of wheel and ladder graphs by utilizing the idea of subdivision.

On the Multiplicative Degree-Based Topological Indices of Silicon-Carbon Si2C3-I[p,q] and Si2C3-II[p,q

The application of graph theory in chemical and molecular structure research has far exceeded people’s expectations, and it has recently grown exponentially. In the molecular graph, atoms are represented by vertices and bonds by edges. Topological indices help us to predict many physico-chemical properties of the concerned molecular compound. In this article, we compute Generalized first and multiplicative Zagreb indices, the multiplicative version of the atomic bond connectivity index, and the Generalized multiplicative Geometric Arithmetic index for silicon-carbon Si2C3−I[p,q] and Si2C3−II[p,q] second.

General multiplicative Zagreb indices of trees

We study trees of given order, given number of pendant vertices, segments and branching vertices. We present upper and lower bounds on the general multiplicative Zagreb indices for trees of given order and number of pendant vertices/segments/branching vertices. Bounds on the first and the second multiplicative Zagreb indices are corollaries of the general results. We also characterize trees having the largest and the smallest indices.

Topological Indices of Vitamin D3

Graph theory has provided chemists with a variety of useful tools, such as topological indices. A topological index Top(G) of a graph G is a number with the property that for every graph H isomorphic to G, Top(H) = Top(G). In this paper, we compute ABC index, ABC4 index, Randi´c connectivity index, Sum connectivity index, GA index , GA5 index, First Zagreb index, Second Zagreb index, First Multiple Zagreb index, Second Multiple Zagreb index, Augmented Zagreb index, Harmonic index and Hyper Zagreb index, First Zagreb polynomial, Second Zagreb polynomial, Third Zagreb polynomial, Forgotten polynomials, Forgotten topological index and Symmetric division index of vitamin D3.

On extremal multiplicative Zagreb indices of trees with given domination number

For a (molecular) graph, the first multiplicative Zagreb index Π1 is equal to the product of squares of the vertex degrees, and the second multiplicative Zagreb index Π2 is equal to the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore the multiplicative Zagreb indices in terms of domination number. Sharp upper and lower bounds of Π1 and Π2 are given. In addition, the corresponding extreme graphs are characterized, and our conclusions enrich and extend some known results.

Zagreb, multiplicative zagreb indices and coindices of 〖nc〗_n (k) and 〖ca〗_3 (c_6 ) graphs

Let be a simple connected graph with vertex set V and edge set E. The first, second and third Zagreb indices of G are defind, respectivly by: , and where is the degree of vertex u in G and uv is an edge of G, connecting the vertices u and v. Recently, the first and second multiplicative Zagreb indices of graph are defind by: and . The first and second Zagreb coindices of graph are defind by: and . and , named as multiplicative Zagreb coindices. In this article, we compute the first, second and the third Zagreb indices and the first and second multiplicative Zagreb indices of some graphs. The first and second Zagreb coindices and the first and second multiplicative Zagreb coindices of these graphs are also computed.

Relations between ordinary and multiplicative degree-based topological indices

Let G be a simple connected graph with n vertices and m edges, and sequence of vertex degrees d1 ? d2 ?...? dn > 0. If vertices i and j are adjacent, we write i ~ j. Denote by ?1, ?*1, Q? and H? the multiplicative Zagreb index, multiplicative sum Zagreb index, general first Zagreb index, and general sumconnectivity index, respectively. These indices are defined as ?1 = ?ni=1 d2i, ?*1 = ?i~j(di+dj), Q? = ?n,i=1 d?i and H? = ?i~j(di+dj)?. We establish upper and lower bounds for the differences H?-m (?1*)?/m and Q?-n(?1)?/2n . In this way we generalize a number of results that were earlier reported in the literature.

On multiplicative Zagreb indices of two new operations of graphs

Recently, Wang et al. (2017) introduced two new operations of graphs. In this paper the upper bounds of the multiplicative Zagreb indices of the two newly proposed operations of graphs are derived.

The Ratio and Product of the Multiplicative Zagreb Indices

‎The first multiplicative Zagreb index $Pi_1(G)$ is equal to the‎ ‎product of squares of the degree of the vertices and the second‎ ‎multiplicative Zagreb index $Pi_2(G)$ is equal to the product of‎ ‎the products of the degree of pairs of adjacent vertices of the‎ ‎underlying molecular graphs $G$‎. ‎Also‎, ‎the multiplicative sum Zagreb index $Pi_3(G)$ is equal to the product of‎ ‎the sums of the degree of pairs of adjacent vertices of $G$‎. ‎In‎ ‎this paper‎, ‎we introduce a new version of the multiplicative sum‎ ‎Zagreb index and study the moments of the ratio and product of ‎all above‎ indices in a randomly chosen molecular graph with tree structure of order $n$. ‏Also, a ‎supermartingale is introduced by ‎‎Doob's supermartingale‎ ‎inequality.

On the bounds of degree-based topological indices of the Cartesian product of F-sum of connected graphs

Topological indices are the mathematical tools that correlate the chemical structure with various physical properties, chemical reactivity or biological activity numerically. A topological index is a function having a set of graphs as its domain and a set of real numbers as its range. In QSAR/QSPR study, a prediction about the bioactivity of chemical compounds is made on the basis of physico-chemical properties and topological indices such as Zagreb, Randić and multiple Zagreb indices. In this paper, we determine the lower and upper bounds of Zagreb indices, the atom-bond connectivity (ABC) index, multiple Zagreb indices, the geometric-arithmetic (GA) index, the forgotten topological index and the Narumi-Katayama index for the Cartesian product of F-sum of connected graphs by using combinatorial inequalities.

Degree-based topological indices of double graphs and strong double graphs

The topological indices are useful tools to the theoretical chemists that are provided by the graph theory. They correlate certain physicochemical properties such as boiling point, strain energy, stability, etc. of chemical compounds. For a graph [Formula: see text], the double graph [Formula: see text] is a graph obtained by taking two copies of graph [Formula: see text] and joining each vertex in one copy with the neighbors of corresponding vertex in another copy and strong double graph SD[Formula: see text] of the graph [Formula: see text] is the graph obtained by taking two copies of the graph [Formula: see text] and joining each vertex [Formula: see text] in one copy with the closed neighborhood of the corresponding vertex in another copy. In this paper, we compute the general sum-connectivity index, general Randi[Formula: see text] index, geometric–arithmetic index, general first Zagreb index, first and second multiplicative Zagreb indices for double graphs and strong double graphs and derive the exact expressions for these degree-base topological indices for double graphs and strong double graphs in terms of corresponding index of original graph [Formula: see text].

Topological Characterization of Carbon Graphite and Crystal Cubic Carbon Structures.

Graph theory is used for modeling, designing, analysis and understanding chemical structures or chemical networks and their properties. The molecular graph is a graph consisting of atoms called vertices and the chemical bond between atoms called edges. In this article, we study the chemical graphs of carbon graphite and crystal structure of cubic carbon. Moreover, we compute and give closed formulas of degree based additive topological indices, namely hyper-Zagreb index, first multiple and second multiple Zagreb indices, and first and second Zagreb polynomials.

Further results on computation of topological indices of certain networks

There are various topological indices such as degree based topological indices, distance based topological indices and counting related topological indices etc. These topological indices correlate certain physicochemical properties such as boiling point, stability of chemical compounds. In this study, the authors compute the sum-connectivity index and multiplicative Zagreb indices for certain networks of chemical importance such as silicate networks, hexagonal networks, oxide networks, and honeycomb networks. Moreover, a comparative study using computer-based graphs has been made to clarify their nature for these families of networks.

On the Certain Topological Indices of Titania Nanotube TiO2[m, n

Abstract A numeric quantity that characterises the whole structure of a molecular graph is called the topological index that predicts the physical features, chemical reactivities, and boiling activities of the involved chemical compound in the molecular graph. In this article, we give new mathematical expressions for the multiple Zagreb indices, the generalised Zagreb index, the fourth version of atom-bond connectivity (ABC4) index, and the fifth version of geometric-arithmetic (GA5) index of TiO2[m, n]. In addition, we compute the latest developed topological index called by Sanskruti index. At the end, a comparison is also included to estimate the efficiency of the computed indices. Our results extended some known conclusions.

On extremal multiplicative Zagreb indices of trees with given number of vertices of maximum degree

The first multiplicative Zagreb index of a graph G is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the products of degrees of pairs of adjacent vertices. In this paper, we explore the trees in terms of given number of vertices of maximum degree. The maximum and minimum values of ∏1(G) and ∏2(G) of trees with arbitrary number of maximum degree are provided. In addition, the corresponding extremal graphs are characterized.

Sharp upper bounds for multiplicative Zagreb indices of bipartite graphs with given diameter

The first multiplicative Zagreb index of a graph G is the product of the square of every vertex degree, while the second multiplicative Zagreb index is the product of the degree of each edge over all edges. In our work, we explore the multiplicative Zagreb indices of bipartite graphs of order n with diameter d, and sharp upper bounds are obtained for these indices of graphs in B(n,d), where B(n,d) is the set of all n-vertex bipartite graphs with the diameter d. In addition, we explore the relationship between the maximal multiplicative Zagreb indices of graphs within B(n,d). As consequences, those bipartite graphs with the largest, second-largest and smallest multiplicative Zagreb indices are characterized, and our results extend and enrich some known conclusions.