Chemical graph theory is a branch of graph theory in which a chemical compound is represented by simple graph called molecular graph in which vertices are atoms of compound and edges are the atomic bounds. A graph is connected if there is at least one connection between its vertices. If a graph does not contain any loop or multiple edge then it is called a network. Between two vertices u and v, the distance is the shortest path between them and is denoted by d(u,v) = dG (u,v) in graph G. For a vertex v of G the “degree” dv is the number of vertices attached with it. The degree and valence in chemistry are closely related with each other. We refer the book  for more details. Now a day another emerging field is Cheminformatics, which helps to predict biological activities with the relationship of Structure-property and quantitative structure-activity. Topological indices and Physico-chemical properties are used in prediction of bioactivity if underlined compounds are used in these studies [7, 8, 9, 10, 11].
A number that describe the topology of a graph is called topological index. In 1947, the first and most studied topological index was introduced by Weiner . For more details about this index can be found in [13, 14]. In 1975, Milan Randić′ introduced the Randić′ index . Bollobas et al.  and Amic et al.  in 1998, working independently defined the generalized Randić′ index. This index was studied by both mathematicians and chemists [18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28].
The first and the second Zagreb indices are defined as
and modified Randić index is defined as
Shigehalli and Kanabur  introduced following new degree-based topological indices: Arithmetic-Geometric (AG1) index
In this report, we aim to compute degree-based topological indices of dominating David derived networks of first type, second type and third type. These networks are constructed and studied in [1, 2, 3, 4, 5].
2 Main results
In this section we present our computational results.
Let D1 (n) be the Dominating David Derived Network of 1st type. Then
SK(D1 (n)) = 297n2 − 341n + 121,
SK2 (D1 (n)) = 1098n2 − 1342n + 490.
In dominating derived network D1 (n) (see Figure 1) there are six type of edges in E(D1 (n)) based on the degree of end vertices, i.e.
It can be observed from Figure 1, that
Let D2 (n) be the dominating David derived network of 2nd type. Then
SK(D2 (n)) = 315n2 − 367n + 131,
SK1 (D2 (n))= 558n2 − 698n + 258,
In dominating derived network D2 (n)(figure 2) there are five type of edges in E(D2 (n)) based on the degree of end vertices. i.e.
It can be observed from figure 2 that
Let D3 (n) be the dominating David derived network of 3rd type. Then
R′(D3 (n)) = 27n2 − 30n + 11,
SK(D3 (n)) = 396n2 − 484n + 176,
SK1 (D3 (n)) = 720n2 − 936n + 352,
SK2 (D3 (n)) = 1476n2 − 1892n + 704.
3 Graphical comparison
In the present report, we computed seven degree-based topological indices of dominating David derived networks of first, second and third type. We compare our results geometrically by plotting computed degree-based indices. We believe that our results play a vital rule in preparation of new drugs.
Imran M., Baig, A.Q., Ali, H. On topological properties of dominating David derived networks, Canad. J. of Chem., 2015, 94(2), 137-148. Google Scholar
Star of David [online]. Available at http://Wikipedia.org/wiki/starofDavid
Deutsch E., Klavzar S., M-Polynomial and degree-based topological indices, Iran. J. Math. Chem., 2015, 6, 93-102. Google Scholar
Trinajstic N., Chemical graph theory, CRC press, 1992 Google Scholar
West D.B., Introduction to graph theory (Vol. 2), Upper Saddle River, Prentice Hall, 2001 Google Scholar
Deng H., Yang J., Xia F., A general modeling of some vertex-degree based topological indices in benzenoid systems and phenylenes, Comp. & Math. with Applic., 2011, 61(10), 3017-3023. CrossrefGoogle Scholar
Gutman I., Polansky O.E., Mathematical concepts in organic chemistry, Springer Science & Business Media, 2012. Google Scholar
Bollobás B., Erdõs P., Graphs of extremal weights, Ars Combinatoria, 1998, 50, 225-233. Google Scholar
Hu Y., Li X., Shi Y., Xu T., Gutman I., On molecular graphs with smallest and greatest zeroth-order general Randić index, MATCH Commun. Math. Comput. Chem, 2005, 54(2), 425-434. Google Scholar
Li X., Gutman I., Mathematical Chemistry Monographs No. 1, Kragujevac, 2006. Google Scholar
Hall L.H., Kier L.B., Molecular connectivity in chemistry and drug research, 1976 Google Scholar
Kier L.B., Hall L.H., Molecular connectivity in structure-activity analysis, Research Studies, 1986. Google Scholar
Li X., Gutman I., Randić M., Mathematical aspects of Randić-type molecular structure descriptors, University, Faculty of Science, 2006. Google Scholar
Randić M., On history of the Randic index and emerging hostility toward chemical graph theory, MATCH Commun. Math. Comput. Chem, 2008, 59, 5-124. Google Scholar
Gutman I., Furtula B., (Eds.) Recent results in the theory of Randic index, University, Faculty of Science, 2008. Google Scholar
Li X., Shi Y., A survey on the Randic index, MATCH Commun. Math. Comput. Chem, 2008, 59(1), 127-156. Google Scholar
Gutmann I., (Ed.) Recent Results in the Theory of Randić Index, Kragujevac University, 2008. Google Scholar
Nikolic S., Kovaèeviæ G., Milièevic A., Trinajstic N., The Zagreb indices 30 years after, Croatica Chemica Acta, 2003, 76(2), 113-124.Google Scholar
Gutman I., Das K.C., The first Zagreb index 30 years after, MATCH Commun. Math. Comput. Chem, 2004, 50, 83-92. Google Scholar
Das K. C., Gutman I., Some properties of the second Zagreb index, MATCH Commun. Math. Comput. Chem, 2004, 52(1), 3-1. Google Scholar
Trinajstic N., Nikolic S., Milièevic A., Gutman I., About the Zagreb Indices, Kemija u industriji, 2010, 59(12), 577-589. Google Scholar
Vukièevic D., Graovac A., Valence connectivity versus Randić, Zagreb and modified Zagreb index: A linear algorithm to check discriminative properties of indices in acyclic molecular graphs, Croatica Chemica Acta, 2004, 77(3), 501-508. Google Scholar
Shigehalli V.S., Kanabur R., Computation of New Degree-Based Topological Indices of Graphene, 2016, 2016, 4341919 Google Scholar
About the article
Published Online: 2017-12-29
Conflict of Interest: The authors declare that there is no conflict of interest regarding the publication of this paper.
Citation Information: Open Physics, Volume 15, Issue 1, Pages 1015–1021, ISSN (Online) 2391-5471, DOI: https://doi.org/10.1515/phys-2017-0126.
© 2017 Muhammad Saeed Ahmad et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License. BY-NC-ND 4.0