Computing First General Zagreb Index of Operations on Graphs

Liu, Jia‐Bao; Javed, Saira; Javaid, Muhammad; Shabbir, Khurram

doi:10.1109/access.2019.2909822

Cited by 50 publications

(41 citation statements)

References 16 publications

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“…So, many chemical graphs can be generated by using simple graphs based on operations such as alkane (C 3 H 6 ) is the corona product of P 3 & N 2 , a type of cycloalkanes is cyclohexane (C 6 H 12 ) that is the corona product of C 6 & N 2 , a polynomial chain and nanotube (TUC 4 (m, n)) are the Cartesian product of P m & P 2 and C m & P 2 , and a fence and closed fence are the lexicographic product of P m & P 2 and C m & P 2 . Up till now, many results of the various TIs have been presented under different molecular graphs based on operations, see [26,[28][29][30][31][32][33][34][35][36][37][38].…”

Section: Introductionmentioning

confidence: 99%

Zagreb Connection Indices of Molecular Graphs Based on Operations

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Topological index (numeric number) is a mathematical coding of the molecular graphs that predicts the physicochemical, biological, toxicological, and structural properties of the chemical compounds that are directly associated with the molecular graphs. The Zagreb connection indices are one of the TIs of the molecular graphs depending upon the connection number (degree of vertices at distance two) appeared in 1972 to compute the total electron energy of the alternant hydrocarbons. But after that, for a long period, these are not studied by researchers. Recently, Ali and Trinajstic Mol. Inform. 372018,1−7 restudied the Zagreb connection indices and reported that the Zagreb connection indices comparatively to the classical Zagreb indices provide the better absolute value of the correlation coefficient for the thirteen physicochemical properties of the octane isomers (all these tested values have been taken from the website http://www.moleculardescriptors.eu). In this paper, we compute the general results in the form of exact formulae & upper bounds of the second Zagreb connection index and modified first Zagreb connection index for the resultant graphs which are obtained by applying operations of corona, Cartesian, and lexicographic product. At the end, some applications of the obtained results for particular chemical structures such as alkanes, cycloalkanes, linear polynomial chain, carbon nanotubes, fence, and closed fence are presented. In addition, a comparison between exact and computed values of the aforesaid Zagreb indices is also included.

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Section: Introductionmentioning

confidence: 99%

Zagreb Connection Indices of Molecular Graphs Based on Operations

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“…In the subject of graph theory, the terms metric dimension, domination number, total domination number, and graph labeling have a vital importance in many real life field problems. In [5], and [26] - [32] discussed the properties ofȃmetric dimension, domination number, total domination number, and graph labeling in detail. Similarly, these concepts are also discussed in fuzzy graph theory [7], [13], [18]- [25] and [33].…”

Section: Introductionmentioning

confidence: 99%

Complex Vague Graphs and Their Application in Decision-Making Problems

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Fuzzy graph models are found everywhere in natural and human made structures, including process dynamics in biological, physical and social systems. Since real-life problems are often uncertain due to inconsistent and indeterminate information, it is very hard for an expert to model those problems using a fuzzy graph. A complex vague model is useful in the field of mathematics which gives more precision, comparability and flexibility to the system as compared to the vague model. In these years, a mathematical approach is a generalized approach of blending different aspects. According to the above mathematical approach, we investigate strong techniques which are based on complex vague graphs. The purpose of this research study is to present and explore the key properties including: Cartesian products, composition, strong product, semi-strong product and direct product of complex vague graphs with examples. Finally, we present application of complex vague graphs in decision-making problems. INDEX TERMS Complex vague graph; Cartesian products; Composition; Strong product; Semi strong product; Direct product.

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“…The extremal graphs with respect to F-index among the unicyclic and bicyclic graphs are studied in [9,10]. For more studies, we refer to [11] and [12][13][14][15][16][17][18][19][20][21][22][23][24][25].…”

Section: Introductionmentioning

confidence: 99%

Bounds on F-index of tricyclic graphs with fixed pendant vertices

2020

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Abstract The F-index F(G) of a graph G is obtained by the sum of cubes of the degrees of all the vertices in G. It is defined in the same paper of 1972 where the first and second Zagreb indices are introduced to study the structure-dependency of total π-electron energy. Recently, Furtula and Gutman [J. Math. Chem. 53 (2015), no. 4, 1184–1190] reinvestigated F-index and proved its various properties. A connected graph with order n and size m, such that m = n + 2, is called a tricyclic graph. In this paper, we characterize the extremal graphs and prove the ordering among the different subfamilies of graphs with respect to F-index in $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$, where $\begin{array}{} \displaystyle {\it\Omega}^{\alpha}_n \end{array}$ is a complete class of tricyclic graphs with three, four, six and seven cycles, such that each graph has α ≥ 1 pendant vertices and n ≥ 16 + α order. Mainly, we prove the bounds (lower and upper) of F(G), i.e $$\begin{array}{} \displaystyle 8n+12\alpha +76\leq F(G)\leq 8(n-1)-7\alpha + (\alpha+6)^3 ~\mbox{for each}~ G\in {\it\Omega}^{\alpha}_n. \end{array}$$

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Computing First General Zagreb Index of Operations on Graphs

Cited by 50 publications

References 16 publications

Zagreb Connection Indices of Molecular Graphs Based on Operations

Zagreb Connection Indices of Molecular Graphs Based on Operations

Complex Vague Graphs and Their Application in Decision-Making Problems

Bounds on F-index of tricyclic graphs with fixed pendant vertices

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