scholarly journals F-Index of some graph operations

2016 ◽  
Vol 08 (02) ◽  
pp. 1650025 ◽  
Author(s):  
Nilanjan De ◽  
Sk. Md. Abu Nayeem ◽  
Anita Pal
Keyword(s):  
Electron Energy ◽  
Topological Index ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Molecular Graphs ◽  
Basic Properties ◽  
Graph Operations ◽  
Physico Chemical ◽  
Vertex Degrees

The F-index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972, in the same paper where the first and second Zagreb indices were introduced to study the structure-dependency of total [Formula: see text]-electron energy. But this topological index was not further studied till then. Very recently, Furtula and Gutman [A forgotten topological index,J. Math. Chem. 53(4) (2015) 1184–1190.] reinvestigated the index and named it “forgotten topological index” or “F-index”. In that paper, they present some basic properties of this index and showed that this index can enhance the physico-chemical applicability of Zagreb index. Here, we study the behavior of this index under several graph operations and apply our results to find the F-index of different chemically interesting molecular graphs and nanostructures.

scholarly journals New Results on the Forgotten Topological Index and Coindex

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Akbar Jahanbani ◽  
Maryam Atapour ◽  
Rana Khoeilar
Keyword(s):  
Electron Energy ◽  
Connected Graph ◽  
Topological Index ◽  
Nonadjacent Vertex ◽  
Zagreb Indices ◽  
Vertex Degrees ◽  
Simple Connected Graph

The ℱ -coindex (forgotten topological coindex) for a simple connected graph G is defined as the sum of the terms ζ G 2 y + ζ G 2 x over all nonadjacent vertex pairs x , y of G , where ζ G y and ζ G x are the degrees of the vertices y and x in G , respectively. The ℱ -index of a graph is defined as the sum of cubes of the vertex degrees of the graph. This was introduced in 1972 in the same paper where the first and second Zagreb indices were introduced to study the structure dependency of total π -electron energy. Therefore, considering the importance of the ℱ -index and ℱ -coindex, in this paper, we study these indices, and we present new bounds for the ℱ -index and ℱ -coindex.

scholarly journals Computing Topological Indices and Polynomials for Line Graphs

Mathematics ◽  
2018 ◽  
Vol 6 (8) ◽  
pp. 137 ◽  
Author(s):  
Shahid Imran ◽  
Muhammad Siddiqui ◽  
Muhammad Imran ◽  
Muhammad Nadeem
Keyword(s):  
Topological Index ◽  
Line Graph ◽  
Topological Indices ◽  
Line Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Second Zagreb Index ◽  
Vertex Degrees ◽  
Set Up ◽  
Ladder 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.

scholarly journals Multiplicative Zagreb Indices of Molecular Graphs

2019 ◽  
Vol 2019 ◽  
pp. 1-19 ◽  
Author(s):  
Xiujun Zhang ◽  
H. M. Awais ◽  
M. Javaid ◽  
Muhammad Kamran Siddiqui
Keyword(s):  
Mathematical Modeling ◽  
Vital Role ◽  
Molecular Structures ◽  
Quantitative Structure ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Molecular Graphs ◽  
Structure Properties

Mathematical modeling with the help of numerical coding of graphs has been used in the different fields of science, especially in chemistry for the studies of the molecular structures. It also plays a vital role in the study of the quantitative structure activities relationship (QSAR) and quantitative structure properties relationship (QSPR) models. Todeshine et al. (2010) and Eliasi et al. (2012) defined two different versions of the 1st multiplicative Zagreb index as ∏Γ=∏p∈VΓdΓp2 and ∏1Γ=∏pq∈EΓdΓp+dΓq, respectively. In the same paper of Todeshine, they also defined the 2nd multiplicative Zagreb index as ∏2Γ=∏pq∈EΓdΓp×dΓq. Recently, Liu et al. [IEEE Access; 7(2019); 105479–-105488] defined the generalized subdivision-related operations of graphs and obtained the generalized F-sum graphs using these operations. They also computed the first and second Zagreb indices of the newly defined generalized F-sum graphs. In this paper, we extend this study and compute the upper bonds of the first multiplicative Zagreb and second multiplicative Zagreb indices of the generalized F-sum graphs. At the end, some particular results as applications of the obtained results for alkane are also included.

scholarly journals Reformulated Zagreb Indices of Some Derived Graphs

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 366 ◽  
Author(s):  
Jia-Bao Liu ◽  
Bahadur Ali ◽  
Muhammad Aslam Malik ◽  
Hafiz Muhammad Afzal Siddiqui ◽  
Muhammad Imran
Keyword(s):  
Physical Properties ◽  
Chemical Structure ◽  
Topological Index ◽  
Chemical Reactivity ◽  
Line Graph ◽  
Total Graph ◽  
Zagreb Indices ◽  
Subdivision Graph ◽  
Chemical Constitution ◽  
Vertex Degrees

A topological index is a numeric quantity that is closely related to the chemical constitution to establish the correlation of its chemical structure with chemical reactivity or physical properties. Miličević reformulated the original Zagreb indices in 2004, replacing vertex degrees by edge degrees. In this paper, we established the expressions for the reformulated Zagreb indices of some derived graphs such as a complement, line graph, subdivision graph, edge-semitotal graph, vertex-semitotal graph, total graph, and paraline graph of a graph.

scholarly journals Note on the Reformulated Zagreb Indices of Two Classes of Graphs

2020 ◽  
Vol 2020 ◽  
pp. 1-4
Author(s):  
Tongkun Qu ◽  
Mengya He ◽  
Shengjin Ji ◽  
Xia Li
Keyword(s):  
Upper Bounds ◽  
Extremal Graphs ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Vertex Degrees

The reformulated Zagreb indices of a graph are obtained from the original Zagreb indices by replacing vertex degrees with edge degrees, where the degree of an edge is taken as the sum of degrees of its two end vertices minus 2. In this paper, we obtain two upper bounds of the first reformulated Zagreb index among all graphs with p pendant vertices and all graphs having key vertices for which they will become trees after deleting their one key vertex. Moreover, the corresponding extremal graphs which attained these bounds are characterized.

scholarly journals Introducing New Exponential Zagreb Indices for Graphs

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Nihat Akgunes ◽  
Busra Aydin
Keyword(s):  
Graph Invariants ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Graph Operations ◽  
General Graphs

New graph invariants, named exponential Zagreb indices, are introduced for more than one type of Zagreb index. After that, in terms of exponential Zagreb indices, lists on equality results over special graphs are presented as well as some new bounds on unicyclic, acyclic, and general graphs are obtained. Moreover, these new graph invariants are determined for some graph operations.

A novel graph invariant: The third leap Zagreb index under several graph operations

2019 ◽  
Vol 11 (05) ◽  
pp. 1950054 ◽  
Author(s):  
Durbar Maji ◽  
Ganesh Ghorai
Keyword(s):  
Tensor Product ◽  
Cartesian Product ◽  
Lexicographic Product ◽  
Graph Invariant ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
The Third ◽  
Graph Operations ◽  
Strong Product ◽  
Corona Product

The third leap Zagreb index of a graph [Formula: see text] is denoted as [Formula: see text] and is defined as [Formula: see text], where [Formula: see text] and [Formula: see text] are the 2-distance degree and the degree of the vertex [Formula: see text] in [Formula: see text], respectively. The first, second and third leap Zagreb indices were introduced by Naji et al. [A. M. Naji, N. D. Soner and I. Gutman, On leap Zagreb indices of graphs, Commun. Combin. Optim. 2(2) (2017) 99–117] in 2017. In this paper, the behavior of the third leap Zagreb index under several graph operations like the Cartesian product, Corona product, neighborhood Corona product, lexicographic product, strong product, tensor product, symmetric difference and disjunction of two graphs is studied.

A Novel Weighted First Zagreb Index of Graph

2020 ◽  
pp. 92-103
Author(s):  
Jibonjyoti Buragohain ◽  
A. Bharali
Keyword(s):  
Connected Graph ◽  
Chemical Properties ◽  
Topological Indices ◽  
First Zagreb Index ◽  
Acentric Factor ◽  
Zagreb Index ◽  
Zagreb Indices ◽  
Physico Chemical ◽  
Octane Isomers ◽  
The First Zagreb Index

The Zagreb indices are the oldest among all degree-based topological indices. For a connected graph G, the first Zagreb index M1(G) is the sum of the term dG(u)+dG(v) corresponding to each edge uv in G, that is, M1 , where dG(u) is degree of the vertex u in G. In this chapter, the authors propose a weighted first Zagreb index and calculate its values for some standard graphs. Also, the authors study its correlations with various physico-chemical properties of octane isomers. It is found that this novel index has strong correlation with acentric factor and entropy of octane isomers as compared to other existing topological indices.

scholarly journals Electron Energy Studying of Molecular Structures via Forgotten Topological Index Computation

2016 ◽  
Vol 2016 ◽  
pp. 1-7 ◽  
Author(s):  
Wei Gao ◽  
Weifan Wang ◽  
Muhammad Kamran Jamil ◽  
Mohammad Reza Farahani
Keyword(s):  
Carbon Atom ◽  
Electron Energy ◽  
Topological Index ◽  
Molecular Graph ◽  
Chemical Properties ◽  
Sum Of Squares ◽  
Molecular Structures ◽  
Physical Chemical ◽  
Vertex Degrees ◽  
Chemical Material

It is found from the earlier studies that the structure-dependency of totalπ-electron energyEπheavily relies on the sum of squares of the vertex degrees of the molecular graph. Hence, it provides a measure of the branching of the carbon-atom skeleton. In recent years, the sum of squares of the vertex degrees of the molecular graph has been defined as forgotten topological index which reflects the structure-dependency of totalπ-electron energyEπand measures the physical-chemical properties of molecular structures. In this paper, in order to research the structure-dependency of totalπ-electron energyEπ, we present the forgotten topological index of some important molecular structures from mathematical standpoint. The formulations we obtained here use the approach of edge set dividing, and the conclusions can be applied in physics, chemical, material, and pharmaceutical engineering.

On Wiener and terminal Wiener index of graphs

2015 ◽  
Vol 08 (05) ◽  
pp. 1550066 ◽  
Author(s):  
J. Baskar Babujee ◽  
J. Senbagamalar
Keyword(s):  
Topological Index ◽  
Wiener Index ◽  
Molecular Graphs ◽  
Structural Descriptor ◽  
Vertex Degrees ◽  
Pendent Vertices ◽  
Sum Of Distances

The Wiener index is a topological index defined as the sum of distances between all pairs of vertices in a graph. It was introduced as a structural descriptor for molecular graphs of alkanes, which are trees with vertex degrees of four at the most. The terminal Wiener index is defined as the sum of distances between all pairs of pendent vertices in a graph. In this paper we investigate Wiener and terminal Wiener for graphs derived from certain operations.

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