Computing Analysis for First Zagreb Connection Index and Coindex of Resultant Graphs

A numeric parameter which studies the behaviour, structural, toxicological, experimental, and physicochemical properties of chemical compounds under several graphs’ isomorphism is known as topological index. In 2018, Ali and Trinajstic studied the first Zagreb connection index to evaluate the value of a molecule. This concept was first studied by Gutman and Trinajstic in 1972 to find the solution of - electron energy of alternant hydrocarbons. In this paper, the first Zagreb connection index and coindex are obtained in the form of exact formulae and upper bounds for the resultant graphs in terms of different indices of their factor graphs, where the resultant graphs are obtained by the product-related operations on graphs such as tensor product, strong product, symmetric difference, and disjunction. At the end, an analysis of the obtained results for the first Zagreb connection index and coindex on the aforesaid resultant graphs is interpreted with the help of numerical values and graphical depictions.

[1]  Dunwei Gong,et al.  Environment Sensitivity-Based Cooperative Co-Evolutionary Algorithms for Dynamic Multi-Objective Optimization , 2018, IEEE/ACM Transactions on Computational Biology and Bioinformatics.

[2]  M. K. Siddiqui,et al.  Multiplicative Zagreb Indices of Molecular Graphs , 2019 .

[3]  Anita Pal,et al.  On Some Bounds and Exact Formulae for Connective Eccentric Indices of Graphs under Some Graph Operations , 2014 .

[4]  Usman Ali,et al.  Zagreb Connection Indices of Molecular Graphs Based on Operations , 2020, Complex..

[5]  Ernesto Estrada,et al.  Generalised topological indices: Optimisation methodology and physico-chemical interpretation , 2005 .

[6]  Jia-Bao Liu,et al.  Computing First General Zagreb Index of Operations on Graphs , 2019, IEEE Access.

[7]  Ali Reza Ashrafi,et al.  The Zagreb coindices of graph operations , 2010, Discret. Appl. Math..

[8]  Muge Togan,et al.  The multiplicative Zagreb indices of graph operations , 2013 .

[9]  Xiaoyan Sun,et al.  Solving interval multi-objective optimization problems using evolutionary algorithms with preference polyhedron , 2011, GECCO '11.

[10]  Svetan M. Ratchev,et al.  Knowledge based requirement engineering for one-of-a-kind complex systems , 2003, Knowl. Based Syst..

[11]  Anita Pal,et al.  The F-coindex of some graph operations , 2016, SpringerPlus.

[12]  Jian-Liang Wu,et al.  Zagreb Eccentricity Indices of the Generalized Hierarchical Product Graphs and Their Applications , 2014, J. Appl. Math..

[13]  Usman Ali,et al.  ZAGREB CONNECTION INDICES OF DISJUNCTION AND SYMMETRIC DIFFERENCE OPERATIONS ON GRAPHS , 2020 .

[14]  H. M. Awais.  Hyper-Zagreb index of graphs based on generalized subdivision related operations , 2020 .

[15]  Jia-Bao Liu,et al.  Computing Zagreb Indices of the Subdivision-Related Generalized Operations of Graphs , 2019, IEEE Access.

[16]  Y. Chu,et al.  On bounds for topological descriptors of φ-sum graphs , 2020, Journal of Taibah University for Science.

[17]  ZAGREB INDICES AND COINDICES OF PRODUCT GRAPHS , 2015 .

[18]  Jianyong Sun,et al.  A novel hybrid multi-objective artificial bee colony algorithm for blocking lot-streaming flow shop scheduling problems , 2018, Knowl. Based Syst..

[19]  K. Das,et al.  On the multiplicative Zagreb coindex of graphs , 2013 .

[20]  I. Gutman,et al.  A forgotten topological index , 2015, Journal of Mathematical Chemistry.

[21]  Usman Ali,et al.  Modified Zagreb connection indices of the T-sum graphs , 2020 .

[22]  Akbar Ali,et al.  First General Zagreb Index of Generalized F-sum Graphs , 2020 .

[23]  Usman Ali.  Upper Bounds of Zagreb Connection Indices of Tensor and Strong Product on Graphs , 2020 .

[24]  Mohammad Reza Farahani,et al.  The hyper-Zagreb index and some graph operations , 2017 .

[25]  Lourdes Santana,et al.  Medicinal chemistry and bioinformatics--current trends in drugs discovery with networks topological indices. , 2007, Current topics in medicinal chemistry.

[26]  N. Trinajstic,et al.  A Novel/Old Modification of the First Zagreb Index , 2017, Molecular informatics.

[27]  I. Gutman,et al.  Graph theory and molecular orbitals. XII. Acyclic polyenes , 1975 .

[28]  I. Gutman,et al.  Beyond the Zagreb indices , 2020, AKCE Int. J. Graphs Comb..

[29]  Usman Ali,et al.  Computing Analysis of Connection-Based Indices and Coindices for Product of Molecular Networks , 2020, Symmetry.

[30]  I. Gutman.  Degree-Based Topological Indices , 2013 .

[31]  Ivan Gutman,et al.  Selected properties of the Schultz molecular topological index , 1994, J. Chem. Inf. Comput. Sci..

[32]  R. Todeschini,et al.  Molecular Descriptors for Chemoinformatics: Volume I: Alphabetical Listing / Volume II: Appendices, References , 2009 .

[33]  Hongbo Hua,et al.  Relations between Zagreb Coindices and Some Distance-Based Topological Indices ∗ , 2012 .

[34]  Muhammad Kamran Siddiqui,et al.  Degree-Based Topological Invariants of Metal-Organic Networks , 2020, IEEE Access.

[36]  Guillaume Ducoffe,et al.  Extremal Graphs with Respect to the Modified First Zagreb Connection Index , 2018, 2018 20th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing (SYNASC).

[37]  Gerta Rücker,et al.  On Topological Indices, Boiling Points, and Cycloalkanes , 1999, J. Chem. Inf. Comput. Sci..

[38]  Usman Ali,et al.  Zagreb Connection Indices of Subdivision and Semi-Total Point Operations on Graphs , 2019 .

[39]  M. Javaid,et al.  Topological properties of metal-organic frameworks , 2020, Main Group Metal Chemistry.

[40]  I. Gutman,et al.  Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons , 1972 .

[41]  On second Zagreb index and coindex of some derived graphs , 2015 .

[42]  M. J. Nikmehr,et al.  The third and hyper-Zagreb coindices of some graph operations , 2016 .

[43]  Ali Iranmanesh,et al.  Chemical Graphs Constructed from Rooted Product and Their Zagreb Indices , 2013 .