Research article

Multiplicative topological properties of graphs derived from honeycomb structure

  • Received: 29 October 2019 Accepted: 16 January 2020 Published: 04 February 2020
  • MSC : 05C12, 05C90

  • Topological indices are numerical parameters of a molecular graph, which characterize its topology and are usually graph invariant. In quantitative structure-activity relationship/quantitative structure-property relationship 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 area of research. In this paper, we are taking Dominating David Derived networks, produced by honeycomb structure of dimension t and obtain analytical closed results of Multiplicative topological indices and acquire exact results of degree based indices.

    Citation: Usman Babar, Haidar Ali, Shahid Hussain Arshad, Umber Sheikh. Multiplicative topological properties of graphs derived from honeycomb structure[J]. AIMS Mathematics, 2020, 5(2): 1562-1587. doi: 10.3934/math.2020107

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  • Topological indices are numerical parameters of a molecular graph, which characterize its topology and are usually graph invariant. In quantitative structure-activity relationship/quantitative structure-property relationship 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 area of research. In this paper, we are taking Dominating David Derived networks, produced by honeycomb structure of dimension t and obtain analytical closed results of Multiplicative topological indices and acquire exact results of degree based indices.


    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. Throughout this paper we take Υ a connected graph. 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 are valuable parameters that are given by graph theory. A number that describe the topology of a graph is called topological index.

    A representation of numbers, polynomials and matrices are representations of a graph. Graph has its own characteristics which can be determined by topological indices and the topology of graph remains unchanged under automorphism of graph. In the different classes of indices, degree based topological indices are of extraordinary significance and assume an essential job in substance graph hypothesis and especially in science. In increasingly exact manner, a topological index Top(H) of a graph, is a number with the property that for each graph G isomorphic to H, Top(H)=Top(G). The idea of topological index originated from Wiener [26] while he was dealing with boiling point of paraffin, named this record as path number. Later on, renamed as Wiener index [7].

    A lot of people have been worked in the Chemical Graph Theory. The importance of honeycomb network can not be ignored in all the work of graph theory. The Honeycomb shape is present everywhere in nature, in plants, animal and human cells. No other shape provides such an optimal cover and strength. It is one of the most stable structures. Honeycomb structures are natural or man-made structures that have the geometry of a honeycomb to allow the minimization of the material used to reach minimal weight and maximum strength. A honeycomb structure provides a material with least density and relative high compression properties and shear properties.

    Built recursively using the hexagon tessellation [24], honeycomb networks are widely used in computer graphics [24], cellular phone base stations, image processing, and in chemistry as the representation of benzenoid hydrocarbons. Honeycomb network HC(t) is obtained from HC(t1) by adding a layer of hexagons around the boundary of HC(t1).

    In this paper, we are going to find the topological indices of graphs derived from the honeycomb structure. Dominating David Derived networks are the graphs derived from honeycomb structure.

    The method of drawing Dominating David Derived networks (dimension t) is as follows.

    STEP 1:-Consider a Honeycomb network HC(t) dimension t, as shown in Figure 1.

    Figure 1.  Construction Algorithm for Dominating David Derived network DDD(2).

    STEP 2:-Split each edge into two by embedding another vertex.

    STEP 3:-In each hexagon cell, connect the new vertices by an edge if they are at a distance of 4 units within the cell.

    STEP 4:-Place vertices at new edge crossings.

    STEP 5:-Remove initial vertices and edges of Honeycomb network.

    STEP 6:-Split each horizontal edge into two edges by inserting a new vertex. The resulting Graph is called Dominating David Derived system DDD(t) of measurement t [22], as shown in Figure 2.

    Figure 2.  Construction Algorithm for Dominating David Derived network DDD(2).

    The First type of Dominating David Derived network D1(t) can be obtained by connecting vertices of degree two by an edge, which are not in the boundary, as shown in Figure 3.

    Figure 3.  First type of Dominating David Derived network D1(2).

    The second type of Dominating David Derived network D2(t) can be obtained by sub dividing once the new edge introduced in D1(t), as shown in Figure 4.

    Figure 4.  Second type of Dominating David Derived network D2(2).

    The Third type of Dominating David Derived network D3(t) can be obtained from D1(t) by introducing parallel path of length 2 between the vertices of degree two which are not in the boundary. See the Figure 5 for third type of Dominating David derived network of dimension 2, D3(2).

    Figure 5.  Third type of Dominating David Derived network D3(2).

    In this article, Υ is considered a network with a V(Υ) vertex set and an edge set of E(Υ), dr is the degree of vertex rV(Υ).

    Some indices associated to Wiener's and Gutman. They derived new topological indices which are named as the first Multiplicative Zagreb index and the second Multiplicative Zagreb index [11] and they are described as:

    Let Υ be a graph. Then

    II1(Υ)=rV(Υ)(dr)2, (1.1)
    II2(Υ)=rsE(Υ)(dr×ds). (1.2)

    V. R. Kulli [15] further described some new and advanced topological indices and he named them as the first Hyper-Zagreb index and the second Hyper-Zagreb index of a graph Υ. They are defined as:

    HII1(Υ)=rsE(Υ)(dr+ds)2, (1.3)
    HII2(Υ)=rsE(Υ)(dr×ds)2. (1.4)

    The first Universal Zagreb index and the second Universal Zagreb index introduced by V. R. Kulli [15]. These indices are defined as:

    MZa1(Υ)=rsE(Υ)(dr+ds)a, (1.5)
    MZa2(Υ)=rsE(Υ)(dr×ds)a. (1.6)

    The sum and product connectivity of Multiplicative indices [15] described as:

    SCII(Υ)=rsE(Υ)1dr+ds, (1.7)
    PCII(Υ)=rsE(Υ)1dr×ds. (1.8)

    Wei Gao et al. [9] define new topological indices which are named as Multiple atom-bond Connectivity index and Multiple Geometric-Arithmetic index and these indices are defined as follow:

    ABCM(Υ)=rsE(Υ)Mr+Ms2Mr×Ms, (1.9)
    GAM(Υ)=rsE(Υ)2Mr×Ms(Mr+Ms), (1.10)

    where

    Mr=rsE(Υ)ds,Ms=rsE(Υ)dr.

    We have study the multiplicative indices such as first and second multiplicative Zagreb Index, first and second hyper-Zagreb index, first and second Universal Zagreb index, sum and product connectivity of multiplicative indices, Multiple atom-bond connectivity index, Multiple Geometric-Arithmetic index and give closed formulae of these indices for Dominating David Derived networks. Haidar et al. studied degree based topological indices for various networks [2]. Nowadays, there is an extensive research activity on ABC and GA indices and their invariants, for further study of topological indices of various graph families see, [1,2,3,4,5,8,12,13,14,16,17,18,19,20,23,27]. For the basic notations and definitions, see [6,21,26].

    In this section, we calculate degree-based topological indices of the dimension t for first type of Dominating David Derived networks. In the coming theorems, we compute some important multiplicative indices.

    Theorem 2.1.1. Consider the first type of Dominating David Derived network Υ1D1(t) for tN. The first and second multiplicative indices of Zagreb indices are equal to

    II1(Υ1)=34560(1+2t)(5+t(13+9t))(4+t(11+9t)),
    II2(Υ1)=339738624t(t1)(7t4)(6+t(9t14))(5+t(9t13))2.

    Proof. Let Υ1 be the first type of Dominating David Derived network. The Υ1 has 20t10 vertices of degree 2, 18t226t+10 vertices of degree 3 and 27t233t+12 vertices of degree 4. The edge set of D1(t) is divided into three partitions based on the degree of end vertices. Table 1, shows such an edge partition of D1(t). Thus from (1.1) is follows that,

    II1(Υ1)=rV(Υ1)(dr)2.
    Table 1.  Edge partition of first type of Dominating David Derived network (D1(t)) based on degrees of end vertices of each edge.
    (dr,ds) where rsE(Υ1) Number of edges
    (2,2) 4t
    (2,3) 4t4
    (2,4) 28t16
    (3,3) 9t213t+5
    (3,4) 36t256t+24
    (4,4) 36t252t+20

     | Show Table
    DownLoad: CSV

    By using vertex partitions, we get

    II1(Υ1)=(2)2(20t10)×(3)2(18t226t+10)×(4)2(27t233t+12),=4(20t10)×9(18t226t+10)×16(27t233t+12),

    By doing some calculations, we have

    II1(Υ1)=34560(1+2t)(5+t(13+9t))(4+t(11+9t)).

    From (1.2), we have

    II2(Υ1)=rsE(Υ1)(dr×ds).

    By using Table 1 edge partitions, we get

    II2(Υ1)=4|E1(Υ1(t))|×6|E2Υ1(t)|×8|E3(Υ1(t))|×9|E4(Υ1(t))|×12|E5(Υ1(t))|×16|E6(Υ1(t))|,=4(4t)×6(4t4)×8(28t16)×9(9t213t+5)×12(36t256t+24)×16(36t252t+20),

    By doing some calculations, we have

    II2(Υ1)=339738624t(t1)(7t4)(6+t(9t14))(5+t(9t13))2.

    Now, we compute advance topological indices and name them as the first Hyper-Zegreb index and second Hyper-Zegreb index for first type of Dominating David Derived network D1(t).

    Theorem 2.1.2. Let Υ1D1(t) be the first type of Dominating David Derived network, then

    HII1(Υ1)=1664719257600t(t1)(7t4)(6+t(9t14))(5+t(9t13))2,
    HII2(Υ1)=112717121716224t(t1)(7t4)(6+t(9t14))(5+t(9t13))2.

    Proof. The outcome can be obtained by using the edge partition in Table 1. By using equation (1.3),

    HII1(Υ1)=rsE(Υ1)(dr+ds)2.
    HII1(Υ1)=16|E1(Υ1(t))|×25|E2(Υ1(t))|×36|E3(Υ1(t))|×36|E4(Υ1(t))|×49|E5(Υ1(t))|×64|E6(Υ1(t))|,=16(4t)×25(4t4)×36(28t16)×36(9t213t+5)×49(36t256t+24)×64(36t252t+20),

    By doing some calculations, we get

    HII1(Υ1)=1664719257600t(t1)(7t4)(6+t(9t14))(5+t(9t13))2.

    Thus from (1.4),

    HII2(Υ1)=rsE(Υ1)(dr×ds)2.
    HII2(Υ1)=16|E1(Υ1(t))|×36|E2(Υ1(t))|×64|E3(Υ1(t))|×81|E4(Υ1(t))|×144|E5(Υ1(t))|×256|E6(Υ1(t))|,=16(4t)×36(4t4)×64(28t16)×81(9t213t+5)×144(36t256t+24)×256(36t252t+20),
    HII2(Υ1)=112717121716224t(t1)(7t4)(6+t(9t14))(5+t(9t13))2.

    Now, we compute the first and second Universal Zagrab indices.

    Theorem 2.1.3. Let Υ1D1(t) be the first type of Dominating David Derived network, thenz

    MZa1(Υ1)=210+7a×315at(t1)(7t4)(6+t(9t14))(5+t(9t13))2,
    MZa2(Υ1)=45+6a×81at(t1)(7t4)(6+t(9t14))(5+t(9t13))2.

    Proof. We get the outcome with the edge partition in Table 1. It follows from (1.5),

    MZa1(Υ1)=rsE(Υ1)(dr+ds)a.
    MZa1(Υ1)=(4)a|E1(Υ1(t))|×(5)a|E2(Υ1(t))|×(6)a|E3(Υ1(t))|×(6)a|E4(Υ1(t))|×(7)a|E5(Υ1(t))|×(8)a|E6(Υ1(t))|,=4a(4t)×5a(4t4)×6a(28t16)×6a(9t213t+5)×7a(36t256t+24)×8a(36t252t+20),

    By doing some calculations, we get

    MZa1(Υ1)=210+7a×315at(t1)(7t4)(6+t(9t14))(5+t(9t13))2.

    Also from (1.6),

    MZa2(Υ1)=rsE(Υ1)(dr×ds)a.
    MZa2(Υ1)=(4)a|E1(Υ1(t))|×(6)a|E2(Υ1(t))|×(8)a|E3(Υ1(t))|×(9)a|E4(Υ1(t))|×(12)a|E5(Υ1(t))|×(16)a|E6(Υ1(t))|,=4a(4t)×6a(4t4)×8a(28t16)×9a(9t213t+5)×12a(36t256t+24)×16a(36t252t+20),

    By making some calculations, we get

    MZa2(Υ1)=45+6a×81at(t1)(7t4)(6+t(9t14))(5+t(9t13))2.

    The sum and product connectivity of multiplicative indices are computed as follows.

    Theorem 2.1.4. Let Υ1D1(t) be the first type of Dominating David Derived network, then

    SCII(Υ1)=643235t(t1)(7t4)(9t214t+6)(9t213t+5)2,
    PCII(Υ1)=169t(t1)(7t4)(9t214t+6)(9t213t+5)2.

    Proof. We get the outcome with the edge partition in Table 1. It follows from (1.7),

    SCII(Υ1)=rsE(Υ1)1dr+ds.
    SCII(Υ1)=12|E1(Υ1(t))|×15|E2(Υ1(t))|×16|E3(Υ1(t))|16|E4(Υ1(t))|×17|E5(Υ1(t))|×18|E6(Υ1(t))|,=12(4t)×15(4t4)×16(28t16)×16(9t213t+5)×17(36t256t+24)×18(36t252t+20),

    By doing some calculations, we get

    SCII(Υ1)=643235t(t1)(7t4)(9t214t+6)(9t213t+5)2.

    Thus from (1.8),

    PCII(Υ1)=rsE(Υ1)1dr×ds.
    PCII(Υ1)=12|E1(Υ1(t))|×16|E2(Υ1(t))|×18|E3(Υ1(t))|×13|E4(Υ1(t))|×112|E5(Υ1(t))|×14|E6(Υ1(t))|,=12(4t)×16(4t4)×18(28t16)+×19(9t213t+5)×112(36t256t+24)×116(36t252t+20),

    By making some calculations, we get

    PCII(Υ1)=169t(t1)(7t4)(9t214t+6)(9t213t+5)2.

    Wei Gao et al. defines topological indices which are named as Multiple atom-bond connectivity index and Multiple Geometric-Arithmetic index and these indices are computed as follows.

    Theorem 2.1.5. Let Υ1D1(t) be the first type of Dominating David Derived network, then

    ABCM(Υ1)=1144(181548171521182912461271974+9941813412165241866206122228249+6330+30396+365703906)+1144t(216+7214+1815+4817+1521+1829+3635+1246+1271+9742194+18134+12165+36186+6206+12222+82495739872570+3906)+1144t2(2794+2398+36570),
    GAM(Υ1)=3737711059762153+202319400677+(488677+416251323813333778295525)t+925(121+503)t2.

    Proof. We get the outcome with the edge partition in Table 2. It follows from (1.9),

    ABCM(Υ1)=rsE(Υ1)Mr+Ms2Mr×Ms.
    ABCM(Υ1)=732|E1(Υ1(t))|+964|E2(Υ1(t))|+35256|E3(Υ1(t))|+67512|E4(Υ1(t))|+17144|E5(Υ1(t))|+37384|E6(Υ1(t))|+31384|E7(Υ1(t))|+55768|E8(Υ1(t))|+23288|E9(Υ1(t))|+831728|E10(Υ1(t))|+471152|E11(Υ1(t))|+551536|E12(Υ1(t))|+712304|E13(Υ1(t))|+291024|E14(Υ1(t))|+953456|E15(Υ1(t))|+1516144|E16(Υ1(t))|+1034608|E17(Υ1(t))|+372048|E18(Υ1(t))|+17512288|E19(Υ1(t))|+151024|E20(Υ1(t))|+19918432|E21(Υ1(t))|,=732(4t)+964(4t)+35256(4)+67512(4t4)+17144(4t4)+37384(4t4)+31384(12t8)+55768(4t4)+23288(2t2)+831728(4t4)+471152(9t27t+3)+551536(4)+712304(4t4)+291024(4t4)+953456(36t272t+36)+1516144(4t4)+1034608(4t4)+372048(4t4)+17512288(4t4)+151024(4t4)+19918432(36t276t+40),
    Table 2.  Edge partition of first type of Dominating David Derived network (D1(t)) based on degrees product of end vertices of each edge.
    (Mr,Ms) where rsE(Υ1) Number of edges (Mr,Ms) where rsE(Υ1) Number of edges
    (8, 8) 4t (48, 64) 4
    (8, 48) 4t (48, 96) 4t-4
    (8, 64) 4 (48,128) 4t-4
    (8,128) 4t-4 (48,144) 36t272t+36
    (12, 24) 4t-4 (48,256) 4t-4
    (12, 64) 4t-4 (64,144) 4t-4
    (16, 48) 12t-8 (96,128) 4t-4
    (16, 96) 4t-4 (96,256) 4t-4
    (24, 24) 2t-2 (128,144) 4t-4
    (24,144) 4t-4 (144,256) 36t276t+40
    (48, 48) 9t27t+3

     | Show Table
    DownLoad: CSV

    By doing some calculations, we get

    ABCM(Υ1)=1144(181548171521182912461271974+9941813412165241866206122228249+6330+30396+365703906)+1144t(216+7214+1815+4817+1521+1829+3635+1246+1271+9742194+18134+12165+36186+6206+12222+82495739872570+3906)+1144t2(2794+2398+36570).

    Thus from (1.10),

    GAM(Υ1)=rsE(Υ1)2Mr×Ms(Mr+Ms).
    GAM(Υ1)=1|E1(Υ1(t))|+267|E2(Υ1(t))|+429|E3(Υ1(t))|+817|E4(Υ1(t))|+223|E5(Υ1(t))|+8319|E6(Υ1(t))|+32|E7(Υ1(t))|+267|E8(Υ1(t))|+1|E9(Υ1(t))|+267|E10(Υ1(t))|+1|E11(Υ1(t))|+437|E12(Υ1(t))|+223|E13(Υ1(t))|+4611|E14(Υ1(t))|+32|E15(Υ1(t))|+12288152|E16(Υ1(t))|+1213|E17(Υ1(t))|+12288112|E18(D1(t))|+24576176|E19(Υ1(t))|+18432136|E20(Υ1(t))|+2425|E21(Υ1(t))|,=1(4t)+267(4t)+429(4)+817(4t4)+223(4t4)+8319(4t4)+32(12t8)+267(4t4)+1(2t2)+267(4t4)+1(9t27t+3)+437(4)+223(4t4)+4611(4t4)+32(36t272t+36)+12288152(4t4)+1213(4t4)+12288112(4t4)+24576176(4t4)+18432136(4t4)+2425(36t276t+40),
    GAM(Υ1)=3737711059762153+202319400677+(488677+416251332381333778295525)t+925(121+503)t2.

    Now, we are calculating certain degree-based multiplicative topological indices of the Υ2D2(t), where tN for second type of Dominating David Derived network.

    Theorem 2.2.1. Consider the second type of Dominating David Derived network Υ2D2(t) for tN. The first and second multiplicative Zagreb indices are equal to

    II1(Υ2)=3456(5+t(13+9t))(4+t(11+9t))(5+t(7+9t)),
    II2(Υ2)=18874368t(7t4)(6+t(9t14))(5+t(9t13))(3+t(9t11)).

    Proof. Let Υ2 be the second type of Dominating David Derived network. The D2(t) has 9t2+7t5 vertices of degree 2, 18t226t+10 vertices of degree 3 and 27t233t+12 vertices of degree 4. The edge set of D2(t) is divided into five partitions based on the degree of end vertices. Table 3, shows such an edge partition of D2(t). Thus from (1.1) is follows that

    II1(Υ2)=rV(Υ2)(dr)2.
    Table 3.  Edge partition of second type of Dominating David Derived network (D2(t)) based on degrees of end vertices of each edge.
    (dr,ds) where rsE(Υ2) Number of edges
    (2,2) 4t
    (2,3) 18t222t+6
    (2,4) 28t16
    (3,4) 36t256t+24
    (4,4) 36t252t+20

     | Show Table
    DownLoad: CSV

    By using vertex partitions, we get

    II1(Υ2)=(2)2(9t2+7t5)×(3)2(18t226t+10)×(4)2(27t233t+12),=4(9t2+7t5)×9(18t226t+10)×16(27t233t+12),

    By doing some calculations, we have

    II1(Υ2)=3456(5+t(13+9t))(4+t(11+9t))(5+t(7+9t)).

    Thus from (1.2), we have

    II2(Υ2)=rsE(Υ2)(dr×ds).

    By using Table 3 edge partitions, we get

    II2(Υ2)=4|E1(Υ2(t))|×6|E2D(Υ2(t))|×8|E3(Υ2(t))|×12|E4(Υ2(t))|×16|E5(Υ2(t))|,=4(4t)×6(18t222t+6)×8(28t16)×12(36t256t+24)×16(36t252t+20),

    By doing some calculations, we have

    II2(Υ2)=18874368t(7t4)(6+t(9t14))(5+t(9t13))(3+t(9t11)).

    Now, we compute the topological indices named as first Hyper-Zagreb index and second Hyper-Zagreb index of a graph Υ2.

    Theorem 2.2.2. Let Υ2D2(t) be the second type of Dominating David Derived network, then

    HII1(Υ2)=23121100800t(7t4)(6+t(9t14))(5+t(9t13))(3+t(9t11)),
    HII2(Υ2)=695784701952t(7t4)(6+t(9t14))(5+t(9t13))(3+t(9t11)).

    Proof. The outcome is obtained by using the edge partition provided in Table 3. The result is from (1.3),

    HII1(Υ2)=rsE(Υ2)(dr+ds)2.
    HII1(Υ2)=16|E1(Υ2(t))|×25|E2(Υ2(t))|×36|E3(Υ2(t))|×49|E4(Υ2(t))|×64|E5(Υ2(t))|,=16(4t)×25(18t222t+6)×36(28t16)×49(36t256t+24)×64(36t252t+20),

    By doing some calculations, we get

    HII1(Υ2)=23121100800t(7t4)(6+t(9t14))(5+t(9t13))(3+t(9t11)).

    Also from (1.4),

    HII2(Υ2)=rsE(Υ2)(dr×ds)2.
    HII2(Υ2)=16|E1(Υ2(t))|×36|E2(Υ2(t))|×64|E3(Υ2(t))|×144|E4(Υ2(t))|×256|E5(Υ2(t))|,=16(4t)+×36(18t222t+6)×64(28t16)×144(36t256t+24)×256(36t252t+20),
    HII2(Υ2)=695784701952t(7t4)(6+t(9t14))(5+t(9t13))(3+t(9t11)).

    Now, we calculate first and second Universal-Zagreb indices.

    Theorem 2.2.3. Let Υ2D2(t) be the second type of Dominating David Derived network, then

    MZa1(Υ2)=83+2a105at(7t4)(6+t(9t14))(5+t(9t13))(3+t(9t11)),
    MZa2(Υ2)=83+4a×9at(7t4)(6+t(9t14))(5+t(9t13))(3+t(9t11)).

    Proof. The outcome is obtained by using the edge partition provided in Table 3. The result is from (1.5),

    MZa1(Υ2)=rsE(Υ2)(dr+ds)a.
    MZa1(Υ2)=(4)a|E1(Υ2(t))|×(5)a|E2(Υ2(t))|×(6)a|E3(Υ2(t))|×(7)a|E4(Υ2(t))|×(8)a|E5(Υ2(t))|,=4a(4t)×5a(18t222t+6)×6a(28t16)×7a(36t256t+24)×8a(36t252t+20),

    By doing some calculations, we get

    MZa1(Υ2)=83+2a105at(7t4)(6+t(9t14))(5+t(9t13))(3+t(9t11)).

    Also from (1.6),

    MZa2(Υ2)=rsE(Υ2)(dr×ds)a.
    MZa2(Υ2)=(4)a|E1(Υ2(t))|×(6)a|E2(Υ2(t))|×(8)a|E3(Υ2(t))|×(12)a|E4(Υ2(t))|×(16)a|E5(Υ2(t))|,=4a(4t)×6a(18t222t+6)×8a(28t16)×12a(36t256t+24)×16a(36t252t+20),
    MZa2(Υ2)=83+4a×9at(7t4)(6+t(9t14))(5+t(9t13))(3+t(9t11)).

    The sum and product connectivity of multiplicative indices are as follows.

    Theorem 2.2.4. Let Υ2D2(t) be the second type of Dominating David Derived network, then

    SCII(Υ2)=64105t(7t4)(9t214t+6)(9t213t+5)(9t211t+3),
    PCII(Υ2)=83t(7t4)(9t214t+6)(9t213t+5)(9t211t+3).

    Proof. The outcome is obtained by using the edge partition provided in Table 3. The result is from (1.7),

    SCII(Υ2)=rsE(Υ2)1dr+ds.
    SCII(Υ2)=12|E1(Υ2(t))|×15|E2(Υ2(t))|×16|E3(Υ2(t))|×17|E4(Υ2(t))|×18|E5(Υ2(t))|,=12(4t)×15(18t222t+6)×16(28t16)×17(36t256t+24)×18(36t252t+20),

    By doing some calculations, we get

    SCII(Υ2)=64105t(7t4)(9t214t+6)(9t213t+5)(9t211t+3).

    Also from (1.8),

    PCII(Υ2)=rsE(Υ2)1dr×ds.
    PCII(Υ2)=12|E1(Υ2(t))|×16|E2(Υ2(t))|×18|E3(Υ2(t))|×112|E4(Υ2(t))|×14|E5(Υ2(t))|,=12(4t)×16(18t222t+6)×18(28t16)×112(36t256t+24)×116(36t252t+20),

    By doing some calculations, we get

    PCII(Υ2)=83t(7t4)(9t214t+6)(9t213t+5)(9t211t+3).

    Now we calculate Multiple atom-bond connectivity index and Multiple Geometric-Arithmetic index.

    Theorem 2.2.5. Let Υ2D2(t) be the second type of Dominating David Derived network, then

    ABCM(Υ2)=14080(1020134252113602351029+1020355104225574+170078+306087510134340165680186170206340222+170330816395+85039872051085906)+14080t(6120+204013+204014+42521+136023+51029+51042+25574442078612087+510134+340165+1020186+170206+340222+8163951615398+720510+85906)+14080t2(306078+306087+765398),
    GAM(Υ2)=(1468845525+18482962690031550313321766385)+t(315728552541409676672+21203133+32326385)+t2(86425+136082451).

    Proof. The outcome is obtained by using the edge partition provided in Table 4. The result is from (1.9),

    ABCM(Υ2)=rsE(Υ2)Mr+Ms2Mr×Ms.
    ABCM(Υ2)=732|E1(Υ2(t))|+964|E2(Υ2(t))|+35256|E3(Υ2(t))|+67512|E4(Υ2(t))|+23144|E5(Υ2(t))|+1396|E6(Υ2(t))|+1396|E7(Υ2(t))|+37384|E8(Υ2(t))|+31384|E9(Υ2(t))|+55768|E10(Υ2(t))|+791152|E11(Υ2(t))|+13256|E12(Υ2(t))|+21512|E13(Υ2(t))|+29768|E14(Υ2(t))|+551536|E15(Υ2(t))|+291024|E16(D2(t))|+1516144|E17(D2(t))|+1034608|E18(Υ2(t))|+372048|E19(Υ2(t))|+17512288|E20(Υ2(t))|+151024|E21(Υ2(t))|+19918432|E22(Υ2(t))|,=732(4t)+964(4t)+35256(4)+67512(4t4)+23144(4t4)+1396(18t230t+14)+1396(4t4)+37384(4t4)+31384(12t8)+55768(4t4)+791152(4t4)+13256(8t4)+21512(4t4)+29768(36t272t+36)+551536(4)+291024(4t4)+1516144(4t4)+1034608(4t4)+372048(4t4)+17512288(4t4)+151024(4t4)+19918432(36t276t+40),
    Table 4.  Edge partition of second type of Dominating David Derived network (D2(t)) based on degrees product of end vertices of each edge.
    (Mr,Ms)whereuvE(Υ2) Number of edges (Mr,Ms)whereuvE(Υ2) Number of edges
    (8, 8) 4t (32, 48) 8t-4
    (8, 48) 4t (32, 96) 4t-4
    (8, 64) 4 (32,144) 36t272t+36
    (8,128) 4t-4 (48, 64) 4
    (9, 16) 4t-4 (48,128) 4t-4
    (9, 32) 18t230t+14 (48,256) 4t-4
    (12, 16) 4t-4 (64,144) 4t-4
    (12, 64) 4t-4 (96,128) 4t-4
    (16, 48) 12t-8 (96,256) 4t-4
    (16, 96) 4t-4 (128,144) 4t-4
    (16,144) 4t-4 (144,256) 36t276t+40

     | Show Table
    DownLoad: CSV

    By making some calculations, we are getting

    ABCM(Υ2)=14080(1020134252113602351029+1020355104225574+170078+306087510134340165680186170206340222+170330816395+85039872051085906)+14080t(6120+204013+204014+42521+136023+51029+51042+25574442078612087+510134+340165+1020186+170206+340222+8163951615398+720510+85906)+14080t2(306078+306087+765398).

    Thus from (1.10),

    GAM(Υ2)=rsE(Υ2)2Mr×Ms(Mr+Ms).
    GAM(Υ2)=1|E1(Υ2(t))|+267|E2(Υ2(t))|+429|E3(Υ2(t))|+817|E4(Υ2(t))|+2425|E5(Υ2(t))|+24241|E6(Υ2(t))|+437|E7(Υ2(t))|+8319|E8(Υ2(t))|+32|E9(Υ2(t))|+267|E10(Υ2(t))|+35|E11(Υ2(t))|+256|E12(Υ2(t))|+32|E13(Υ2(t))|+6211|E14(Υ2(t))|+437|E15(Υ2(t))|+4611|E16(Υ2(t))|+12288152|E17(Υ2(t))|+1213|E18(Υ2(t))|+12288112|E19(Υ2(t))|+24576176|E20(Υ2(t))|+18432136|E21(Υ2(t))|+2425|E22(Υ2(t))|,=1(4t)+267(4t)+429(4)+817(4t4)+2425(4t4)+24241(18t230t+14)+437(4t4)+8319(4t4)+32(12t8)+267(4t4)+35(4t4)+265(8t4)+32(4t4)+6112(36t272t+36)+437(4)+4611(4t4)+12288152(4t4)+1213(4t4)+12288112(4t4)+24576176(4t4)+18432136(4t4)+2425(36t276t+40),
    GAM(Υ2)=(1468845525+18482962690031550313321766385)+t(315728552541409627667+21203133+32326385)+t2(86425+136082451).

    In this section, we calculate the multiplicative topological degree based indices for third type of Dominating David Derived network D3(t) of dimension t.

    Theorem 2.3.1. Consider the third type of Dominating David Derived network Υ3D3(t) for tN. The first and second multiplicative Zagreb indices are equal to

    II1(Υ3)=384t(1+3t)(22+t(59+45t)),
    II2(Υ3)=32768t2(9t5)(11+9t(2t3)).

    Proof. Let Υ3 be the third type of Dominating David Derived network. The Υ3 has 18t26t vertices of degree 2 and 45t259t+22 vertices of degree 4. The edge set of D3(t) is divided into three partitions based on the degrees of end vertices. Table 5, shows such an edge partition of D3(t). Thus from (1.1) is follows that,

    II1(Υ3)=rV(Υ3)(dr)2.
    Table 5.  Edge partition of third type of Dominating David Derived network (D3(t)) based on degrees of end vertices of each edge.
    (dr,ds) where rsE(Υ3) Number of edges
    (2,2) 4t
    (2,4) 36t220t
    (4,4) 72t2108t+44

     | Show Table
    DownLoad: CSV

    By using vertex partition, we get

    II1(Υ3)=(2)2(18t26t)×(4)2(45t259t+22),II1(Υ3)=4(18t26t)×16(45t259t+22),

    By making some calculations, we have

    II1(Υ3)=384t(1+3t)(22+t(59+45t)).

    Also from (1.2), we have

    II2(Υ3)=rsE(Υ3)(dr×ds).

    By using Table 5 edge partitions, we get

    II2(Υ3)=4|E1(Υ3(t))|×8|E2Υ3(t)|×16|E3Υ3(t)|,=4(4t)×8(36t220t)×16(72t2108t+44),

    By making some calculations, we have

    II2(Υ3)=32768t2(9t5)(11+9t(2t3)).

    Now we calculate some advanced topological indices named as first Hyper-Zagreb index and second Hyper-Zagreb index of a graph Υ3.

    Theorem 2.3.2. Let Υ3D3(t) be the third type of Dominating David Derived network, then

    HII1(Υ3)=2359296t2(9t5)(11+9t(2t3)),
    HII2(Υ3)=16777216t2(9t5)(11+9t(2t3)).

    Proof. The outcome is obtained by using the edge partition provided in Table 5. The result is from (1.3)

    HII1(Υ3)=pqE(Υ3)(dp+dq)2.
    HII1(Υ3)=16|E1(Υ2(t))|×36|E2(Υ2(t))|×64|E3(Υ2(t))|,=16(4t)×36(36t220t)×64(72t2108t+44),

    By doing some calculations, we get

    HII1(Υ3)=2359296t2(9t5)(11+9t(2t3)).

    Also from (1.4),

    HII2(Υ3)=pqE(Υ3)(dp×dq)2.
    HII2(Υ3)=16|E1(Υ3(t))|×64|E2(Υ3(t))|×256|E3(Υ3(t))|,
    HII2(Υ3)=16(4t)×64(36t220t)×256(72t2108t+44),

    By doing some calculations, we get

    HII2(Υ3)=16777216t2(9t5)(11+9t(2t3)).

    Now the first and second Universal-Zagreb indices are defined as.

    Theorem 2.3.3. Let Υ3D3(t) be the third type of Dominating David Derived network, then

    MZa1(Υ3)=3a641+at2(9t5)(11+9t(2t3)),
    MZa2(Υ3)=82+3at2(9t5)(11+9t(2t3)).

    Proof. The outcome is obtained by using the edge partition provided in Table 5. The result is from (1.5),

    MZa1(Υ3)=pqE(G)(dp+dq)a.
    MZa1(Υ3)=(4)a|E1(Υ3(t))|×(6)a|E2(Υ3(t))|×(8)a|E3(Υ3(t))|,=4a(4t)×6a(36t220t)×8a(72t2108t+44),

    By doing some calculation, we get

    MZa1(Υ3)=3a641+at2(9t5)(11+9t(2t3)).

    Thus from (1.6),

    MZa2(Υ3)=pqE(Υ3)(dp×dq)a.
    MZa2(Υ3)=(4)a|E1(Υ3(t))|×(8)a|E2(Υ3(t))|×(16)a|E3(Υ3(t))|,=4a(4t)×8a(36t220t)×16a(72t2108t+44),
    MZa2(Υ3)=82+3at2(9t5)(11+9t(2t3)).

    The sum and product connectivity of multiplicative indices are described as follows.

    Theorem 2.3.4. Let Υ3D3(t) be the third type of Dominating David Derived network of type 3, then

    SCII(Υ3)=83t2(9t5)(11+9t(2t3)),
    PCII(Υ3)=22t2(9t5)(11+9t(2t3)).

    Proof. The outcome is obtained by using the edge partition provided in Table 5. The result is from (1.7),

    SCII(Υ3)=uvE(G)1dp+dq.
    SCII(Υ3)=12|E1(Υ3(t))|×16|E2(Υ3(t))|×18|E3(Υ3(t))|,=12(4t)×16(36t220t)×18(72t2108t+44),

    By doing some calculation, we get

    SCII(Υ3)=83t2(9t5)(11+9t(2t3)).

    Also from (1.8),

    PCII(Υ3)=pqE(Υ3)1dp×dq.
    PCII(Υ3)=12|E1(Υ3(t))|×18|E2(Υ3(t))|×14|E3(Υ3(t))|,=12(4t)×18(36t220t)×116(72t2108t+44),

    By doing some calculations we get

    PCII(Υ3)=22t2(9t5)(11+9t(2t3)).

    The Multiple atom-bond connectivity index and Multiple Geometric-Arithmetic index are calculated as follows.

    Theorem 2.3.5. Let Υ3D3(t) be the third type of Dominating David Derived network, then

    ABCM(Υ3)=1192(14423+48352471+9678249524134121436254+24318+30510+32573)+1192t(16814+14423+4835+247126478+2495+24134+12143+62549031857510+32573)+1192t2(21678+54318+27510),
    GAM(Υ3)=(36t260t+36)+817(4t4)+892(4t4)+1092(4t+4)+232(8t8)+232(12t12)+45(36t260t+16)+45(36t244t+16).

    Proof. The outcome is obtained by using the edge partition provided in Table 6. The result is from (1.9).

    Table 6.  Edge partition of third type of Dominating David Derived network (D3(t)) based on degrees product of end vertices of each edge.
    (Mr,Ms) where rsE(Υ3) Number of edges (Mr,Ms) where rsE(Υ3) Number of edges
    (8, 8) 4t (64, 64) 8t
    (8, 64) 4t+4 (64,128) 8t-8
    (8,128) 4t-4 (64,256) 36t260t+16
    (16, 32) 12t-12 (128,128) 4t-4
    (16, 64) 36t244t+16 (128,256) 4t+4
    (16,128) 4t-4 (256,256) 36t276t+40
    (32,256) 4t-4

     | Show Table
    DownLoad: CSV
    ABCM(Υ3)=rsE(Υ3)Mr+Ms2Mr×Ms.
    ABCM(Υ3)=732|E1(Υ3(t))|+35256|E2(Υ3(t))|+67512|E3(Υ3(t))|+23256|E4(Υ3(t))|+39512|E5(Υ3(t))|+711024|E6(Υ3(t))|+1434096|E7(Υ3(t))|+632048|E8(Υ3(t))|+954096|E9(Υ3(t))|+1598192|E10(Υ3(t))|+1278192|E11(Υ3(t))|+19116384|E12(Υ3(t))|+25532768|E13(Υ3(t))|,=732(4t)+35256(4t+4)+67512(4t4)+23256(12t12)+39512(36t244t+16)+711024(4t4)+1434096(4t4)+632048(8t)+954096(8t8)+1598192(36t260t+16)+1278192(4t4)+19116384(4t+4)+25532768(36t276t+40),

    By doing some calculations, we get

    ABCM(Υ3)=1192(14423+48352471+9678249524134121436254+24318+30510+32573)+1192t(16814+14423+4835+247126478+2495+24134+12143+62549031857510+32573)+1192t2(21678+54318+27510).

    Also from (1.10),

    GAM(Υ3)=rsE(Υ3)2Mr×Ms(Mr+Ms).
    GAM(Υ3)=(1)|E1(Υ3(t))|+429|E2(Υ3(t))|+817|E3(Υ3(t))|+223|E4(Υ3(t))|+45|E5(Υ3(t))|+429|E6(Υ3(t))|+8192144|E7(Υ3(t))|+(1)|E8(Υ3(t))|+819296|E9(Υ3(t))|+45|E10(Υ3(t))|+(1)|E11(Υ3(t))|+(32768192)|E12(Υ3(t))|+(1)|E13(Υ3(t))|,=(1)|(4t)+429(4t+4)+817(4t4)|+223(12t12)+45(36t244t+16)+429(4t4)+8192144(4t4)+(1)(8t)+819296(8t8)+45(36t260t+16)+(1)(4t4)+32768192(4t+4)+(1)(36t276t+40),
    GAM(Υ3)=(36t260t+36)+817(4t4)+892(4t4)+1092(4t+4)+232(8t8)+232(12t12)+45(36t260t+16)+45(36t244t+16).

    The graphical representations of topological indices of these networks are depicted in Figures 6 and 7 for certain values of t. By varying the different values of t, the graphs are increasing. These graphs show the accuracy of the results.

    Figure 6.  Comparison of ABCM index for γ1, γ2 and γ3.
    Figure 7.  Comparison of GAM index for γ1, γ2 and γ3.

    In this article, we computed degree-based indices for some derived graphs of HCn graph. We also computed certain degree-based polynomials such as Multiplicative Zagreb, Hyper Zagreb, Universal Zagreb, Sum and Product connectivity of Multiplicative indices, Multiple Atom-Bond connectivity index and Multiple Geometric Arithmetic index for three types of Dominating David Derived networks. We also gave index comparison of these networks. Almost all indices increase with increase in t. These facts may be useful for people working in computer science and chemistry who encounter honeycomb networks. Finding expressions of derived graphs like these is an open problem for many other topological indices.

    The authors are very grateful to the referees for their careful reading with corrections and useful comments, which improved this work very much

    The authors declare no conflict of interest.



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