Canadian Journal of Chemistry. On Topological Properties of Dominating David Derived Networks

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1 Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks Journal: Canadian Journal of Chemistry Manuscript ID cjc r1 Manuscript Type: Article Date Submitted by the Author: 9-Sep-015 Complete List of Authors: Imran, Muhammad; National University of Sciences & Technology (NUST), School of Natural Sciences Baig, A. Q.; COMSATS Insitute of Information Technology, Attock, Pakistan, Mathematics Ali, Haidar; COMSATS Insitute of Information Technology, Attock, Pakistan, Mathematics Keyword: General Randi\'{c} index, Atom-bond connectivity $(ABC)$ index, Geometric-arithmetic $(GA)$ index, Star of David, Dominating David Derived network

2 Page 1 of Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks Muhammad Imran, Abdul Qudair Baig 1, Haidar Ali Department of Mathematics, School of Natural Sciences (SNS), National University of Sciences and Technology (NUST), Sector H-1, Islamabad, Pakistan Tel: imrandhab@gmail.com 1 Department of Mathematics, COMSATS Institute of Information Technology, Attock Department of Mathematics, Government College University, Faisalabad {aqbaig1, haidar0}@gmail.com This research is supported by COMSATS Attock and NUST, Islamabad, Pakistan and by the grant of Higher Education Commission of Pakistan Ref. No. 0-67/NRPU/R&D/HEC/1/1.

3 Canadian Journal of Chemistry Page of M. Imran, A. Q. Baig, H. Ali Abstract. Topological indices are numerical parameters of a graph which characterize its molecular topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as Randić, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this important area of research. All the studied interconnection networks in this paper are constructed by the Star of David network. In this paper, we study the general Randić, first Zagreb, ABC, GA, ABC and GA 5 indices for the first, second and third type of Dominating David Derived networks and give closed formulas of these indices for these networks. These results are useful in network science to understand the underlying topologies of these networks. 010 Mathematics Subject Classification: 05C1, 05C90 Keywords: General Randić index, Atom-bond connectivity (ABC) index, Geometric-arithmetic (GA) index, Star of David, Dominating David Derived network.

4 Page of Canadian Journal of Chemistry 1 Introduction and preliminary results On Topological Properties of Dominating David Derived Networks Graph theory has provided chemists with a variety of useful tools, such as topological indices. Molecules and molecular compounds are often modeled by a molecular graph. A molecular graph is a representation of the structural formula of a chemical compound in terms of graph theory, whose vertices correspond to the atoms of the compound and edges correspond to chemical bonds. Cheminformatics is new subject which is a combination of chemistry, mathematics and information science. It studies Quantitative structure-activity (QSAR) and structure-property (QSPR) relationships that are used to predict the biological activities and properties of the chemical compounds. In the QSAR /QSPR study, physicochemical properties and topological indices such as Wiener index, Szeged index, Randić index, Zagreb indices and ABC index are used to predict bioactivity of the chemical compounds. A graph can be recognized by a numeric number, a polynomial, a sequence of numbers or by a matrix. A topological index is a numeric quantity associated with a graph which characterize the topology of graph and is invariant under graph automorphism. There are some major classes of topological indices such as distance based topological indices, degree based topological indices and counting related polynomials and indices of graphs. Among these classes degree based topological indices are of great importance and play a vital role in chemical graph theory and particularly in theoretical chemistry. In more precise way, a topological index T op(g) of a graph, is a number with the property that for every graph H isomorphic to G, we have T op(h) = T op(g). The concept of topological indices came from the work done by Wiener 1 while he was working on the boiling point of paraffin (an important member of alkane family). He named this index as path number. Later on, the path number was renamed as Wiener index and the whole theory of topological indices started. The algorithm for constructing David Derived and Dominating David Derived network (of dimension n) is as follows: Consider a Star of David network SD(n) of dimension n. Split each edge of it into two by inserting a new vertex. Resulting graph is called David Derived network DD(n) of dimension n. Consider a honeycomb network denoted by HC(n) of dimension n. Split each edge of HC(n) into two by inserting a new vertex. In each hexagon cell, connect the new vertices by an edge if they are at a distance of units within the cell. Place vertices at new edge crossings. Remove the initial vertices and edges of HC(n). Split each horizontal edge into two edges by inserting a new vertex. Resulting graph is called Dominating David Derived network DDD(n) of dimension n. The Dominating David Derived network of first type denoted by D 1 (n) of dimension n can be obtained by connecting vertices of degree two of DDD(n) by an edge, which are not in the boundary. The Dominating David Derived network of second type denoted by D (n) of dimension n can be obtained by subdividing once the new edge introduced in D 1 (n). The Dominating David Derived network of third type D (n) of dimension n can be obtained from D 1 (n) by introducing parallel path of length between the vertices of degree two which are not in the boundary.

5 Canadian Journal of Chemistry Page of M. Imran, A. Q. Baig, H. Ali In this article, G is considered to be a network with vertex set V (G) and edge set E(G), the deg(u) is the degree of vertex u V (G) and S u = deg(v) where N G (u) = {v V (G) uv E(G)}. The v N G (u) notations used in this article are mainly taken from the books 5,6. Let G be a connected graph. Then the Wiener index of G is defined as W (G) = 1 d(u, v) (1) (u,v) where (u, v) is any ordered pair of vertices in G and d(u, v) is u v geodesic. The very first and oldest degree based topological index is Randić index 7 denoted by R 1 (G) and was introduced by Milan Randić and is defined as R 1 (G) = 1 deg(u)deg(v) () The general Randić index R α (G) is the sum of (deg(u)deg(v)) α over all the edges e = uv E(G) and is defined as R α (G) = (deg(u)deg(v)) α for α = 1, 1, 1, 1 () An important topological index introduced by Ivan Gutman and Trinajstić is the Zagreb index denoted by M 1 (G) and is defined as M 1 (G) = (deg(u) + deg(v)) () One of the well-known degree based topological index is atom-bond connectivity (ABC) index introduced by Estrada et al. and defined as ABC(G) = deg(u) + deg(v) deg(u)deg(v) (5) Another well-known connectivity topological descriptor is geometric-arithmetic (GA) index which was introduced by Vukičević et al. 9 and defined as GA(G) = deg(u)deg(v) deg(u) + deg(v) (6) The fourth version of atom-bond connectivity index denoted by ABC and fifth version of geometricarithmetic index denoted by GA 5 can be computed if we are able to find the edge partition of these interconnection networks based on sum of the degrees of end vertices of each edge in these graphs. The

6 Page 5 of Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks 5 fourth version of ABC index is introduced by Ghorbani et al. 10 and defined as ABC (G) = Su + S v S u S v (7) Recently, the fifth version of GA index is proposed by Graovac et al. 11 and is defined as GA 5 (G) = S u S v S u + S v () The general Randić index for α = 1 is the second Zagreb index for any graph G. Main results In this paper, we study the general Randić, first Zagreb, ABC, GA, ABC and GA 5 indices and give closed formulas of these indices for the first, second and third type of Dominating David Derived network. Hayat et al. studied recently the degree based topological indices for various networks like silicates, hexagonal, honeycomb and oxide 1. Nowadays there is an extensive research activity on ABC and GA indices and their variants. For further study of topological indices of various graph families and their invariants see, 1..1 Results for Dominating David Derived networks of first type In this section, we compute certain degree based topological indices of Dominating David Derived network of first type denoted by D 1 (n). We compute the general Randić R α (G) for α = {1, 1, 1, 1 }, ABC, GA, ABC and GA 5 indices for first type of Dominating David Derived networks in this section. In the following theorem, the general Randić index for Dominating David Derived network of first type is computed. Theorem.1.1. Let G = D 1 (n) be the Dominating David Derived network of first type, then its general Randić index is equal to 109n 157n + 501, α = 1; ( )n + ( )n , α = 1 R α (D 1 (n)) = ; 1 (5n n ), α = 1; (6 + 1)n + ( )n , α = 1.

7 Canadian Journal of Chemistry Page 6 of 6 M. Imran, A. Q. Baig, H. Ali Fig. 1. Dominating David Derived network of first type (D 1 ()) (d u, d v ) where uv E(G) Number of edges (, ) n (, ) n (, ) n (, ) 9n 1n + 5 (, ) 6n 56n + (, ) 6n 5n + 0 Table 1. Edge partition of Dominating David Derived network of first type (D 1 (n)) based on degrees of end vertices of each edge. Proof. Let G be the Dominating David Derived network of first type. The Dominating David Derived network of first type (D 1 (n)) has 0n 10 vertices of degree, 1n 6n + 10 vertices of degree and 7n n+1 vertices of degree. The edge set of D 1 (n) can be divided into six partitions based on the degree of end vertices. The first edge partition E 1 (D 1 (n)) contains n edges uv; where deg(u) = deg(v) =. The second edge partition E (D 1 (n)) contains n edges uv, where deg(u) = and deg(v) =. The third edge partition E (D 1 (n)) contains n edges uv; where deg(u) = and deg(v) =. The fourth edge partition E (D 1 (n)) contains 9n 1n + 5 edges uv; where deg(u) = deg(v) =. The fifth edge partition E 5 (D 1 (n)) contains 6n 56n + edges uv; where we have deg(u) = and deg(v) =. The sixth edge partition E 6 (D 1 (n)) contains 6n 5n + 0 edges uv; where deg(u) = deg(v) =. Table 1 shows such an edge partition of D 1 (n). Thus from equation (), is follows that R α (G) = () α For α = 1 Now we apply the formula of general Randić index R α (G) for α = 1. 6 R 1 (G) =

8 Page 7 of Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks 7 By using the edge partition given in Table 1, we get the following R 1 (G) = E 1 (D 1 (n)) + 6 E (D 1 (n)) + E (D 1 (n)) + 9 E (D 1 (n)) + 1 E 5 (D 1 (n)) + E 6 (D 1 (n)) = R 1 (G) = 109n 157n For α = 1 We apply the formula of R α (G) for α = 1. R 1 (G) = 6 By using the edge partition given in Table 1, we get R 1 (G) = E 1(D 1 (n)) + 6 E (D 1 (n)) + E (D 1 (n)) + E (D 1 (n)) + E 5 (D 1 (n)) + E 6 (D 1 (n)) = R 1 (G) = ( )n + ( )n For α = 1 We apply the formula of R α (G) for α = 1. R 1 (G) = 6 1 R 1 (G) = 1 E 1(D 1 (n)) E (D 1 (n)) + 1 E (D 1 (n)) E (D 1 (n)) E 5(D 1 (n)) + 1 E 6(D 1 (n)) = R 1 (G) = 1 (5n n ) For α = 1 We apply the formula of R α (G) for α = 1. R 1 (G) = 6 1 R 1 (G) = 1 E 1(D 1 (n)) E (D 1 (n)) + E (D 1 (n)) + 1 E (D 1 (n)) + 6 E 5(D 1 (n)) + 1 E 6(D 1 (n)) = R 1 (G) = (6 + 1)n + ( )n In the following theorem, we compute the first Zagreb index for Dominating David Derived network of first type denoted by D 1 (n).

9 Canadian Journal of Chemistry Page of M. Imran, A. Q. Baig, H. Ali Theorem.1.. For Dominating David Derived network of first type (G = D 1 (n), the first Zagreb index is equal to M 1 (D 1 (n)) = 59n 6n + Proof. Let G be the Dominating David Derived network of first type (D 1 (n)). By using the edge partition from Table 1, the result directly follows. From the equation (), we have M 1 (G) = 6 (deg(u) + deg(v)) = (deg(u) + deg(v)) M 1 (G) = E 1 (D 1 (n)) + 5 E (D 1 (n)) + 6 E (D 1 (n)) + 6 E (D 1 (n)) + 7 E 5 (D 1 (n)) + E 6 (D 1 (n)). By doing some elementary calculation, we get = M 1 (G) = 59n 6n + In the next theorem, we compute the ABC, GA, ABC and GA 5 indices of Dominating David Derived network of first type denoted by D 1 (n). Theorem.1.. Let G = D 1 (n) be the Dominating David Derived network of first type for every positive integer n ; then we have ABC(D 1 (n)) = ( )n + ( )n + ( ). GA(D 1 (n)) = ( )n + ( )n + ( ABC (D 1 (n)) = ( )n + ( GA 5 (D 1 (n)) = ( )n + ( )n ) )n Proof. By finding the edge partition given in Table 1 and then by using the definition, we get the required result. From equation (5), it follows that ABC(G) = deg(u) + deg(v) = 6 deg(u) + deg(v) ABC(D 1 (n)) = 1 E 1 (D 1 (n)) + 1 E (D 1 (n)) + 1 E (D 1 (n)) + E (D 1 (n)) E 5(D 1 (n)) + 6 E 6(D 1 (n)).

10 Page 9 of Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks 9 By doing the some calculation, we get = ABC(D 1 (n)) = ( )n + ( )n + ( ). From equation(6), we get GA(D 1 (n)) = By doing some calculation, we get deg(u) + deg(v) = 6 deg(u) + deg(v) GA(D 1 (n)) = E 1 (D 1 (n)) E (D 1 (n)) + E (D 1 (n)) + E (D 1 (n)) + 7 E 5(D 1 (n)) + E 6 (D 1 (n)) = GA(D 1 (n)) = ( )n + ( )n + ( ). If we consider edge partitions based on degree sum of neighbors of end vertices; then the edge set E(D 1 (n)) can be divided into twenty edge partition E j (D 1 (n)), 7 j 6, where the edge partition E 7 (D 1 (n)) contains n edges uv with S u = S v = 6; the edge partition E (D 1 (n)) contains n edges uv with S u = 6 and S v = 11; the edge partition E 9 (D 1 (n)) contains edges uv with S u = 6 and S v = 1; the edge partition E 10 (D 1 (n)) contains n edges uv with S u = 6 and S v = 1; the edge partition E 11 (D 1 (n)) contains n edges uv with S u = 7 and S v = 9; the edge partition E 1 (D 1 (n)) contains n edges uv with S u = 7 and S v = 1; the edge partition E 1 (D 1 (n)) contains 1n edges uv with S u = and S v = 11; the edge partition E 1 (D 1 (n)) contains n edges uv with S u = and S v = 1; the edge partition E 15 (D 1 (n)) contains n edges uv with S u = S v = 9; the edge partition E (D 1 (n)) contains n edges uv with S u = 9 and S v = 1; the edge partition E 17 (D 1 (n)) contains 9n 7n + edges uv with S u = S v = 11; the edge partition E 1 (D 1 (n)) contains edges uv with S u = 11 and S v = 1; the edge partition E 19 (D 1 (n)) contains n edges uv with S u = 11 and S v = 1; the edge partition E 0 (D 1 (n)) contains 6n 6n + edges uv with S u = 11 and S v = 1; the edge partition E 1 (D 1 (n)) contains n edges uv with S u = 11 and S v = ; the edge partition E (D 1 (n)) contains n edges uv with S u = 1 and S v = 1; the edge partition E (D 1 (n)) contains n edges uv with S u = 1 and S v = 1; the edge partition E (D 1 (n)) contains n edges uv with S u = 1 and S v = ; the edge partition E 5 (D 1 (n)) contains n edges uv with S u = S v = 1 and the edge partition E 6 (D 1 (n)) contains 6n 76n + 0 edges uv with S u = 1 and S v =. From equation (7), we get ABC (G) = Su + S v S u S v = 6 j=7 uv E j(g) By using the edge partition given in Table, we get the following Su + S v S u S v. ABC (D 1 (n)) = 10 6 E 7(D 1 (n)) E (D 1 (n)) + E 9(D 1 (n)) + 1 E 10(D 1 (n)) + E 11(D 1 (n)) + 57 E 1(D 1 (n)) + 7 E 1(D 1 (n)) E 1(D 1 (n)) + 9 E 15(D 1 (n)) E (D 1 (n)) +

11 Canadian Journal of Chemistry Page 10 of 10 M. Imran, A. Q. Baig, H. Ali (S u, S v ) where uv E(G) Number of edges (S u, S v ) where uv E(G) Number of edges (6, 6) n (11, 11) 9n 7n + (6, 11) n (11, 1) (6, 1) (11, 1) n (6, 1) n (11, 1) 6n 6n + (7, 9) n (11, ) n (7, 1) n (1, 1) n (, 11) 1n (1, 1) n (, 1) n (1, ) n (9, 9) n (1, 1) n (9, 1) n (1, ) 6n 76n + 0 Table. Edge partition of Dominating David Derived network of first type (D 1 (n)) based on sum of degrees of end vertices of each edge E 17(D 1 (n)) + 77 E 1(D 1 (n)) E 19(D 1 (n)) + 15 E 0 (D 1 (n)) E 1(D 1 (n)) E (D 1 (n)) E (D 1 (n)) E (D 1 (n)) E 5(D 1 (n)) + E 6(D 1 (n)). After some calculation, we get the following = ABC (D 1 (n)) = ( )n +( From equation(), we get )n GA 5 (G) = S u S v S u + S v = 6 j=7 uv E j(g) S u S v S u + S v. By using the edge partition given in Table, we get the following GA 5 (D 1 (n)) = E 7 (D 1 (n)) E (D 1 (n)) + E 9(D 1 (n)) E 10(D 1 (n)) + 7 E 11(D 1 (n)) E 1(D 1 (n)) + 19 E 1(D 1 (n)) E 1(D 1 (n)) + E 15 (D 1 (n)) +6 1 E (D 1 (n)) + E 17 (D 1 (n)) + E 1(D 1 (n)) E 19(D 1 (n)) E 0(D 1 (n)) E 1(D 1 (n)) + 1 E (D 1 (n)) E (D 1 (n)) E (D 1 (n)) + E 5 (D 1 (n)) E 6(D 1 (n)). After some calculations, we have = GA 5 (D 1 (n)) = ( )n + ( )n

12 Page 11 of Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks 11. Results for Dominating David Derived networks of second type In this section, we calculate some degree based topological indices of Dominating David Derived network of second type denoted by D (n) of dimension n. We compute the general Randić index R α (G) with α = {1, 1, 1, 1 }, ABC, GA, ABC and GA 5 for D (n). Theorem..1. Let G = D (n) denotes the Dominating David Derived network of second type, then its general Randić index is equal to 11n 196n + 5, α = 1; ( )n + ( )n , α = 1 R α (D (n)) = 10n 5n+7 1, α = 1; ( )n + (7 6 11)n , α = 1. Fig.. Dominating David Derived network of second type (D ()) (d u, d v ) where uv E(G) Number of edges (, ) n (, ) 1n n + 6 (, ) n (, ) 6n 56n + (, ) 6n 5n + 0 Table. Edge partition of Dominating David Derived network of second type (D (n)) based on degrees of end vertices of each edge. Proof. Let G be the Dominating David Derived network of second type. The second Dominating David Derived network denoted by D (n) has 9n + 7n 5 vertices of degree, 1n 6n + 10 vertices of

13 Canadian Journal of Chemistry Page 1 of 1 M. Imran, A. Q. Baig, H. Ali degree and 7n n + 1 vertices of degree. The edge set of D (n) is divided into five partitions based on the degree of end vertices. The first edge partition E 1 (D (n)) contains n edges uv; where deg(u) = deg(v) =. The second edge partition E (D (n)) contains 1n n + 6 edges uv; where deg(u) = and deg(v) =. The third edge partition E (D (n)) contains n edges uv; where deg(u) = and deg(v) =. The fourth edge partition E (D (n)) contains 6n 56n + edges uv; where deg(u) = and deg(v) =. The fifth edge partition E 5 (D (n)) contains 6n 5n + 0 edges uv; where deg(u) = deg(v) =. Table shows such an edge partition of Dominating David Derived network of second type (D (n)). Thus from equation (), it follows that R α (G) = () α For α = 1 Now we apply the formula of general Randić index R α (G) for α = 1. 5 R 1 (G) = By using the edge partition given in Table, we get R 1 (G) = E 1 (D (n)) + 6 E (D (n)) + E (D (n)) + 1 E (D (n)) + E 5 (D (n)). = R 1 (G) = 11n 196n + 5. For α = 1 We apply the formula of R α (G) for α = 1. R 1 (G) = 5 By using the edge partition given in Table, we get R 1 (D (n)) = E 1 (D (n)) + 6 E (D (n)) + E (D (n)) + E (D (n)) + E 5 (D (n)). = R 1 (D (n)) = ( )n + ( )n For α = 1 We apply the formula of R α (G) for α = 1. R 1 (G) = 5 1

14 Page 1 of Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks 1 R 1 (D (n)) = 1 E 1(D (n)) E (D (n)) + 1 E (D (n)) E (D (n)) + 1 E 5(D (n)) = R 1 (D (n)) = 10n 5n For α = 1 We apply the formula of R α (G) for α = 1. R 1 (G) = 5 1 R 1 (D (n)) = 1 E 1(D (n)) E (D (n)) + E (D (n)) + 6 E (D (n)) + 1 E 5(D (n)) = R 1 (D (n)) = ( )n + (7 11)n In the following theorem, we compute the first Zagreb index of Dominating David Derived network of second type denoted by D (n). Theorem... For the Dominating David Derived network of second type G = D (n), the first Zagreb index is equal to M 1 (D (n)) = 60n 7n + 6 Proof. Let D (n) denotes the second Dominating David Derived network of second type. By using the edge partition from Table, the result follows. From equation (), we have M 1 (G) = 5 (deg(u) + deg(v)) = (deg(u) + deg(v)) M 1 (D (n)) = E 1 (D (n)) + 5 E (D (n)) + 6 E (D (n)) + 7 E (D (n)) + E 5 (D (n)) By doing the some calculation, we get the following = M 1 (D (n)) = 60n 7n + 6 Next, we compute the ABC, GA, ABC and GA 5 indices of Dominating David Derived network of second type denoted by D (n). Theorem... Let G = D (n) be the Dominating David Derived network of second type, then we have ABC(G) = ( )n + ( )n , for every positive integer n 1. GA(G) = ( )n + ( )n , for every

15 Canadian Journal of Chemistry Page 1 of 1 M. Imran, A. Q. Baig, H. Ali positive integer n 1. ABC (G) = ( )n + ( )n , for every positive integer n. GA 5 (G) = ( )n + ( )n , for every positive integer n. (S u, S v ) where uv E(G) Number of edges (S u, S v ) where uv E(G) Number of edges (6, 6) n (10, 11) n (6, ) n (10, 1) n (6, 10) 1n 0n + 1 (10, 1) 6n 7n + 6 (6, 11) n (11, 1) (6, 1) (11, 1) n (6, 1) n (11, ) n (7, ) n (1, 1) n (7, 1) n (1, 1) n (, 11) 1n (1, ) n (, 1) n (1, 1) n (, 1) n (1, ) 6n 76n + 0 Table. Edge partition of Dominating David Derived network of second type (D (n)) based on sum of degrees of end vertices of each edge. Proof. By using the edge partition given in Table, we get the result. From equation (5), it follows that ABC(G) = deg(u) + deg(v) = 5 deg(u) + deg(v). ABC(D (n)) = 1 E 1 (D (n)) + 1 E (D (n)) + 1 E (D (n)) E (D (n)) + 6 E 5(D (n)). By doing the some calculation, we get = ABC(D (n))=( )n + ( )n From equation(6), we get GA(D (n)) = deg(u) + deg(v) = 5 deg(u) + deg(v).

16 Page 15 of Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks 15 By using the edge partition given in Table, we get GA(D (n)) = E 1 (D (n)) E (D (n)) + 6 E (D (n)) + 7 E (D (n)) + E 5 (D (n)). By doing the some calculation, we get = GA(D (n))=( )n + ( )n If we consider edge partitions based on degree sum of neighbors of end vertices, then the edge set E(D (n)) can be divided into twenty two edge partition E j (D (n)), 6 j 7; where the edge partition E 6 (D (n)) contains n edges uv with S u = S v = 6. The edge partition E 7 (D (n)) contains n edges uv with S u = 6 and S v =, the edge partition E (D (n)) contains 1n 0n + 1 edges uv with S u = 6 and S v = 10, the edge partition E 9 (D (n)) contains n edges uv with S u = 6 and S v = 11; the edge partition E 10 (D (n)) contains edges uv with S u = 6 and S v = 1; the edge partition E 11 (D (n)) contains n edges uv with S u = 6 and S v = 1; the edge partition E 1 (D (n)) contains n edges uv with S u = 7 and S v = ; the edge partition E 1 (D (n)) contains n edges uv with S u = 7 and S v = 1; the edge partition E 1 (D (n)) contains 1n edges uv with S u = and S v = 11; the edge partition E 15 (D (n)) contains n edges uv with S u = and S v = 1, the edge partition E (D (n)) contains n edges uv with S u = and S v = 1, the edge partition E 17 (D (n)) contains n edges uv with S u = 10 and S v = 11; the edge partition E 1 (D (n)) contains n edges uv with S u = 10 and S v = 1; the edge partition E 19 (D (n)) contains 6n 7n+6 edges uv with S u = 10 and S v = 1, the edge partition E 0 (D (n)) contains edges uv with S u = 11 and S v = 1, the edge partition E 1 (D (n)) contains n edges uv with S u = 11 and S v = 1; the edge partition E (D (n)) contains n edges uv with S u = 11 and S v = ; the edge partition E (D (n)) contains n edges uv with S u = 1 and S v = 1; the edge partition E (D (n)) contains n edges uv with S u = 1 and S v = 1; the edge partition E 5 (D (n)) contains n edges uv with S u = 1 and S v = ; the edge partition E 6 (D (n)) contains n edges uv with S u = S v = 1 and the edge partition E 7 (D (n)) contains 6n 76n + 0 edges uv with S u = 1 and S v =. From equation (7), we get ABC (G) = Su + S v S u S v = 7 j=6 uv E j(g) By using the edge partition given in Table, we get the following Su + S v S u S v. ABC (D (n))= 10 6 E 6(D (n)) + 1 E 7(D (n)) E (D (n)) E 9(D (n)) + E 10(D (n)) + 1 E 11(D (n)) + 1 E 1(D (n)) + 57 E 1(D (n)) + 7 E 1(D (n)) E 15(D (n)) E 19 (D (n)) E 1 17(D (n)) + 10 E 1(D (n)) E 19(D (n)) + 77 E 0(D (n)) +

17 Canadian Journal of Chemistry Page of M. Imran, A. Q. Baig, H. Ali 15 E 1(D (n)) E (D (n)) E (D (n)) E (D (n)) E 5(D (n)) E 6(D (n)) + E 7(D (n)). After some calculation, we get the following = ABC (D (n)) = ( )n + ( )n From equation (), we get GA 5 (G) = S u S v S u + S v = 7 j=6 uv E j(g) By using the edge partition given in Table, we get the following S u S v S u + S v. GA 5 (D (n))= E 6 (D (n)) + 7 E 7(D (n)) + 15 E (D (n)) E 9(D (n)) + E 10(D (n)) E 11(D (n)) E 1(D (n)) E 1(D (n)) + 19 E 1(D (n)) E 15(D (n)) E (D (n)) E 17(D (n)) + 10 E 1(D (n)) E 19(D (n)) + E 0(D (n)) E 1(D (n)) E (D (n)) + 1 E (D (n)) E (D (n)) E 5(D (n)) + E 6 (D (n)) E 7(D (n)). After calculation, we get = GA 5 (D (n)) = ( )n + ( )n Results for Dominating David Derived networks of third type We calculate certain degree based topological indices of Dominating David Derived network of third type denoted by D (n) (of dimension n) in this section. We compute the general Randić index R α (G) with α = {1, 1, 1, 1 }, ABC, GA, ABC and GA 5 in the following theorems for D (n). The general Randić index for D (n) is computed in the following theorem.

18 Page 17 of Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks 17 Theorem..1. Let G = D (n) denotes the Dominating David Derived network of third type, then its general Randić index is equal to 10n 17n + 70, α = 1; ( + 7 )n (0 + )n + 176, α = 1 R α (D (n)) = 6n n+11, α = 1; (1 + 9 )n (5 + 5 )n +, α = 1. Fig.. Dominating David Derived network of third type type (D ()) (d u, d v ) where uv E(G) Number of edges (, ) n (, ) 6n 0n (, ) 7n 10n + Table 5. Edge partition of Dominating David Derived network of third type (D (n)) based on degrees of end vertices of each edge. Proof. Let D (n) be the Dominating David Derived network of third type. The Dominating David Derived network of third type denoted by D (n) has 1n 6n vertices of degree and 5n 59n + vertices of degree. The edge set of D (n) is divided into three partitions based on the degree of end vertices. The first edge partition E 1 (D (n)) contains n edges uv; where deg(u) = deg(v) =. The second edge partition E (D (n)) contains 6n 0n edges uv; where deg(u) = and deg(v) =. The third edge partition E (D (n)) contains 7n 10n + edges uv; where deg(u) = deg(v) =. Table 5 shows such an edge partition of D (n). Thus from equation (), is follows that R α (G) = () α

19 Canadian Journal of Chemistry Page 1 of 1 M. Imran, A. Q. Baig, H. Ali For α = 1 Now we apply the formula of R α (G) for α = 1. R 1 (G) = By using the edge partition given in Table 5, we get R 1 (G) = E 1 (D (n)) + E (D (n)) + E (D (n)) = R 1 (G) = 10n 17n + 70 For α = 1 We apply the formula of R α (G) for α = 1. R 1 (G) = By using the edge partition given in Table 5, we get R 1 (G) = E 1(D (n)) + E (D (n)) + E (D (n)) = R 1 (G) = ( + 7 )n (0 + )n For α = 1 We apply the formula of R α (G) for α = 1. R 1 (G) = 1 R 1 (G) = 1 E 1(D (n)) + 1 E (D (n)) + 1 E (D (n)) = R 1 (G) = 6n n + 11 For α = 1

20 Page 19 of Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks 19 We apply the formula of R α (G) for α = 1. R 1 (G) = 1 R 1 (G) = 1 E 1(D (n)) + E (D (n)) + 1 E (D (n)) = R 1 (G) = (1 + 9 )n (5 + 5 )n + In the following theorem, we compute the first Zagreb index of Dominating David Derived network of third type denoted by D (n). Theorem... For Dominating David Derived network of third type denoted by G = D (n), the first Zagreb index is equal to M 1 (D (n)) = 79n 96n + 5. Proof. Let D (n) denotes the Dominating David Derived network of third type. By using the edge partition from Table 5, the result follows. From equation() we have M 1 (G) = (deg(u) + deg(v)) = (deg(u) + deg(v)) M 1 (G) = E 1 (D (n)) + 6 E (D (n)) + E (D (n)) By doing some calculation, we get = M 1 (G) = 79n 96n + 5 Next, we compute the ABC, GA, ABC and GA 5 indices for Dominating David Derived network of third type denoted by D (n). Theorem... Let G = D (n) be the Dominating David Derived network of third type, then we have ABC(G) = ( )n ( + 7 6)n , for every positive integer n 1. GA(G) = (7 + )n ( )n +, for every positive integer n 1. ABC (G) = ( )n + ( )n , for every positive integer n. GA 5 (G) = ( )n + ( )n , for every positive integer n.

21 Canadian Journal of Chemistry Page 0 of 0 M. Imran, A. Q. Baig, H. Ali (S u, S v ) where uv E(G) Number of edges (S u, S v ) where uv E(G) Number of edges (6, 6) n (1, 1) n (6, 1) n + (1, 1) n (6, 1) n (1, ) 6n 60n + (, 10) 1n 1 (1, 1) n (, 1) 6n n + (1, ) n + (, 1) n (, ) 6n 76n + 0 (10, ) n Table 6. Edge partition of Dominating David Derived network of third type(d (n) )based on sum of degrees of end vertices of each edge. Proof. By using the edge partition given in Table 5, we get the result. From equation (5), it follows that ABC(G) = deg(u) + deg(v) = deg(u) + deg(v). ABC(D (n)) = 1 E 1 (D (n)) E (D (n)) + E (D (n)). By doing some calculation, we get = ABC(G) = ( )n ( + 7 6)n From equation (6), we get GA(G) = deg(u) + deg(v) = deg(u) + deg(v). By doing some calculation, we get By doing some calculation, we get GA(D (n)) = E 1 (D (n)) + E (D (n)) + E (D (n)). = GA = (7 + )n ( )n +. If we consider edge partitions based on degree sum of neighbors of end vertices, then the edge set E(D (n)) can be divided into thirteen edge partition E j (D (n)), j ; where the edge partition E (D (n)) contains n edges uv with S u = S v = 6; the edge partition E 5 (D (n)) contains n + edges uv with S u = 6 and S v = 1; the edge partition E 6 (D (n)) contains n edges uv with S u = 6 and S v = 1; the edge partition E 7 (D (n)) contains 1n 1 edges uv with S u = and S v = 10, the edge partition E (D (n)) contains 6n n + edges uv with S u = and S v = 1, the edge partition E 9 (D (n)) contains n edges uv with S u = and S v = 1; the edge partition E 10 (D (n)) contains

22 Page 1 of Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks 1 n edges uv with S u = 10 and S v = ; the edge partition E 11 (D (n)) contains n edges uv with S u = S v = 1; the edge partition E 1 (D (n)) contains n edges uv with S u = 1 and S v = 1; the edge partition E 1 (D (n)) contains 6n 60n + edges uv with S u = 1 and S v = ; the edge partition E 1 (D (n)) contains n edges uv with S u = S v = 1; the edge partition E 15 (D (n)) contains n + edges uv with S u = 1 and S v = ; and the edge partition E (D (n)) contains 6n 76n + 0 edges uv with S u = S v =. From equation (7), we get ABC (G) = Su + S v S u S v = j= uv E j(g) Su + S v S u S v. By using the edge partition given in Table 6, we get the following ABC (D (n))= 10 6 E (D (n)) + E 5(D (n)) + 1 E 6(D (n)) E 7(D (n)) + E (D (n)) E 9(D (n)) E 10(D (n)) + 1 E 11(D (n)) E 1(D (n)) + 7 E 1(D (n)) E 1(D (n)) + E 15(D (n)) + 0 E (D (n)). After some calculation, we get = ABC (D (n)) = ( )n + ( )n From equation (), we get GA 5 (G) = S u S v S u + S v = By using the edge partition given in Table 6, we have j= uv E j(g) S u S v S u + S v. GA 5 (D (n)))= E (D (n)) + E 5(D (n)) E 6(D (n)) E 7(D (n)) E (D (n)) E 9(D (n)) E 10(D (n)) + E 11 (D (n)) + 1 E 1(D (n)) + 7 E 1(D (n)) + E 1 (D (n)) E 15(D (n)) + E (D (n)). After calculation, we get = GA 5 (D (n))) = ( )n + ( )n

23 Canadian Journal of Chemistry Page of M. Imran, A. Q. Baig, H. Ali Conclusion In this paper, certain degree based topological indices, namely general Randić index, atom-bond connectivity index (ABC), geometric-arithmetic index (GA) and first Zagreb index for Dominating David Derived networks of first, second and third type were studied for the first time and analytical closed formulas for these networks were determined which will help the people working in network science to understand and explore the underlying topologies of these networks. In future, we are interested to design some new architectures/networks and then study their topological indices which will be quite helpful to understand their underlying topologies. References 1. Wiener H. J. Am. Chem. Soc. 197, 69, 17.. Deza, M.; Fowler, P. W.; Rassat, A.; Rogers, K. M. J. Chem. Inf. Comput. Sci. 000, 0, Star of David [online] available, of David.. Simonraj, F.; George, A. GRAPH-HOC 01,, Diudea, M. V.; Gutman, I.; Lorentz, J. Molecular Topology; Nova Science Publishers: Hauppauge, NY, Gutman, I.; Polansky, O. E. Mathematical concepts in organic chemistry, Springer-Verlag, New York, Randić, M. J. Amer. Chem. Soc. 1975, 97, Estrada, E.; Torres, L.; Rodríguez, L.; Gutman, I. Indian J. Chem. 199, 7A, Vukičević, D.; Furtula, B. J. Math. Chem. 009, 6, Ghorbani, M.; Hosseinzadeh, M. A. Optoelectron. Adv. Mater. Rapid Commun. 010,, Graovac, A.; Ghorbani, M.; Hosseinzadeh, M. A. J. Math. Nanosci. 011, 1,. 1. Hayat, S.; Imran, M. Appl. Math. Comput. 01, 0, Bača, M.; Horváthová, J.; Mokrišová, M.; Suhányiovǎ, A. Appl. Math. Comput. 015, 51, Baig, A. Q.; Imran, M.; Ali, H. Optoelectron. Adv. Mater. Rapid Commun. 015, 9,. 15. Baig, A. Q.; Imran, M.; Ali, H. Canad. J. Chem. 015, 9, 70.. Hayat, S.; Imran, M. J. Comput. Theor. Nanosci. 015, 1, Hayat, S.; Imran, M. J. Comput. Theor. Nanosci. 015, 1, Hayat, S.; Imran, M. Appl. Math. Comput. 01, 0, Iranmanesh, A.; Zeraatkar, M. Optoelectron. Adv. Mater. Rapid Commun. 010,, Lin, W.; Chen, J.; Chen, Q.; Gao, T.; Lin, X.; Cai, B. MATCH Commun. Math. Comput. Chem.01, 7, Manuel, P. D.; Abd-El-Barr, M. I.; Rajasingh, I.; Rajan, B. J. Discrete Algorithms 00, 6, 11.. Palacios, J.L. MATCH Commun. Math. Comput. Chem. 01, 7, Simonraj, F.; George, A. Int. J. Future Comput. Commun., 01,, 90.

24 Page of Canadian Journal of Chemistry On Topological Properties of Dominating David Derived Networks Figure Captions Fig. 1. Dominating David Derived network of first type (D 1 ()) Fig.. Dominating David Derived network of second type (D ()) Fig.. Dominating David Derived network of third type type (D ())