
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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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(t−1) by adding a layer of hexagons around the boundary of HC(t−1).
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.
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.
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.
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.
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).
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 r∈V(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)=∏r∈V(Υ)(dr)2, | (1.1) |
II2(Υ)=∏rs∈E(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)=∏rs∈E(Υ)(dr+ds)2, | (1.3) |
HII2(Υ)=∏rs∈E(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)=∏rs∈E(Υ)(dr+ds)a, | (1.5) |
MZa2(Υ)=∏rs∈E(Υ)(dr×ds)a. | (1.6) |
The sum and product connectivity of Multiplicative indices [15] described as:
SCII(Υ)=∏rs∈E(Υ)1√dr+ds, | (1.7) |
PCII(Υ)=∏rs∈E(Υ)1√dr×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)=∑rs∈E(Υ)√Mr+Ms−2Mr×Ms, | (1.9) |
GAM(Υ)=∑rs∈E(Υ)2√Mr×Ms(Mr+Ms), | (1.10) |
where
Mr=∏rs∈E(Υ)ds,Ms=∏rs∈E(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 Υ1≅D1(t) for t∈N. 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(t−1)(7t−4)(6+t(9t−14))(5+t(9t−13))2. |
Proof. Let Υ1 be the first type of Dominating David Derived network. The Υ1 has 20t−10 vertices of degree 2, 18t2−26t+10 vertices of degree 3 and 27t2−33t+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)=∏r∈V(Υ1)(dr)2. |
(dr,ds) where rs∈E(Υ1) | Number of edges |
(2,2) | 4t |
(2,3) | 4t−4 |
(2,4) | 28t−16 |
(3,3) | 9t2−13t+5 |
(3,4) | 36t2−56t+24 |
(4,4) | 36t2−52t+20 |
By using vertex partitions, we get
II1(Υ1)=(2)2(20t−10)×(3)2(18t2−26t+10)×(4)2(27t2−33t+12),=4(20t−10)×9(18t2−26t+10)×16(27t2−33t+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)=∏rs∈E(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(4t−4)×8(28t−16)×9(9t2−13t+5)×12(36t2−56t+24)×16(36t2−52t+20), |
By doing some calculations, we have
⟹II2(Υ1)=339738624t(t−1)(7t−4)(6+t(9t−14))(5+t(9t−13))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 Υ1≅D1(t) be the first type of Dominating David Derived network, then
HII1(Υ1)=1664719257600t(t−1)(7t−4)(6+t(9t−14))(5+t(9t−13))2, |
HII2(Υ1)=112717121716224t(t−1)(7t−4)(6+t(9t−14))(5+t(9t−13))2. |
Proof. The outcome can be obtained by using the edge partition in Table 1. By using equation (1.3),
HII1(Υ1)=∏rs∈E(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(4t−4)×36(28t−16)×36(9t2−13t+5)×49(36t2−56t+24)×64(36t2−52t+20), |
By doing some calculations, we get
⟹HII1(Υ1)=1664719257600t(t−1)(7t−4)(6+t(9t−14))(5+t(9t−13))2. |
Thus from (1.4),
HII2(Υ1)=∏rs∈E(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(4t−4)×64(28t−16)×81(9t2−13t+5)×144(36t2−56t+24)×256(36t2−52t+20), |
⟹HII2(Υ1)=112717121716224t(t−1)(7t−4)(6+t(9t−14))(5+t(9t−13))2. |
Now, we compute the first and second Universal Zagrab indices.
Theorem 2.1.3. Let Υ1≅D1(t) be the first type of Dominating David Derived network, thenz
MZa1(Υ1)=210+7a×315at(t−1)(7t−4)(6+t(9t−14))(5+t(9t−13))2, |
MZa2(Υ1)=45+6a×81at(t−1)(7t−4)(6+t(9t−14))(5+t(9t−13))2. |
Proof. We get the outcome with the edge partition in Table 1. It follows from (1.5),
MZa1(Υ1)=∏rs∈E(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(4t−4)×6a(28t−16)×6a(9t2−13t+5)×7a(36t2−56t+24)×8a(36t2−52t+20), |
By doing some calculations, we get
⟹MZa1(Υ1)=210+7a×315at(t−1)(7t−4)(6+t(9t−14))(5+t(9t−13))2. |
Also from (1.6),
MZa2(Υ1)=∏rs∈E(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(4t−4)×8a(28t−16)×9a(9t2−13t+5)×12a(36t2−56t+24)×16a(36t2−52t+20), |
By making some calculations, we get
⟹MZa2(Υ1)=45+6a×81at(t−1)(7t−4)(6+t(9t−14))(5+t(9t−13))2. |
The sum and product connectivity of multiplicative indices are computed as follows.
Theorem 2.1.4. Let Υ1≅D1(t) be the first type of Dominating David Derived network, then
SCII(Υ1)=643√235t(t−1)(7t−4)(9t2−14t+6)(9t2−13t+5)2, |
PCII(Υ1)=169t(t−1)(7t−4)(9t2−14t+6)(9t2−13t+5)2. |
Proof. We get the outcome with the edge partition in Table 1. It follows from (1.7),
SCII(Υ1)=∏rs∈E(Υ1)1√dr+ds. |
SCII(Υ1)=12|E1(Υ1(t))|×1√5|E2(Υ1(t))|×1√6|E3(Υ1(t))|1√6|E4(Υ1(t))|×1√7|E5(Υ1(t))|×1√8|E6(Υ1(t))|,=12(4t)×1√5(4t−4)×1√6(28t−16)×1√6(9t2−13t+5)×1√7(36t2−56t+24)×1√8(36t2−52t+20), |
By doing some calculations, we get
⟹SCII(Υ1)=643√235t(t−1)(7t−4)(9t2−14t+6)(9t2−13t+5)2. |
Thus from (1.8),
PCII(Υ1)=∏rs∈E(Υ1)1√dr×ds. |
PCII(Υ1)=12|E1(Υ1(t))|×1√6|E2(Υ1(t))|×1√8|E3(Υ1(t))|×13|E4(Υ1(t))|×1√12|E5(Υ1(t))|×14|E6(Υ1(t))|,=12(4t)×1√6(4t−4)×1√8(28t−16)+×1√9(9t2−13t+5)×1√12(36t2−56t+24)×1√16(36t2−52t+20), |
By making some calculations, we get
⟹PCII(Υ1)=169t(t−1)(7t−4)(9t2−14t+6)(9t2−13t+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 Υ1≅D1(t) be the first type of Dominating David Derived network, then
ABCM(Υ1)=1144(−18√15−48√17−15√21−18√29−12√46−12√71−9√74+9√94−18√134−12√165−24√186−6√206−12√222−8√249+6√330+30√396+36√570−3√906)+1144t(216+72√14+18√15+48√17+15√21+18√29+36√35+12√46+12√71+9√74−21√94+18√134+12√165+36√186+6√206+12√222+8√249−57√398−72√570+3√906)+1144t2(27√94+2√398+36√570), |
GAM(Υ1)=373771105−976√2153+202√319−400√677+(488√677+416√251−3238133√3−3778295525)t+925(121+50√3)t2. |
Proof. We get the outcome with the edge partition in Table 2. It follows from (1.9),
ABCM(Υ1)=∑rs∈E(Υ1)√Mr+Ms−2Mr×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(4t−4)+√17144(4t−4)+√37384(4t−4)+√31384(12t−8)+√55768(4t−4)+√23288(2t−2)+√831728(4t−4)+√471152(9t2−7t+3)+√551536(4)+√712304(4t−4)+√291024(4t−4)+√953456(36t2−72t+36)+√1516144(4t−4)+√1034608(4t−4)+√372048(4t−4)+√17512288(4t−4)+√151024(4t−4)+√19918432(36t2−76t+40), |
(Mr,Ms) where rs∈E(Υ1) | Number of edges | (Mr,Ms) where rs∈E(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) | 36t2−72t+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) | 36t2−76t+40 |
(48, 48) | 9t2−7t+3 |
By doing some calculations, we get
⟹ABCM(Υ1)=1144(−18√15−48√17−15√21−18√29−12√46−12√71−9√74+9√94−18√134−12√165−24√186−6√206−12√222−8√249+6√330+30√396+36√570−3√906)+1144t(216+72√14+18√15+48√17+15√21+18√29+36√35+12√46+12√71+9√74−21√94+18√134+12√165+36√186+6√206+12√222+8√249−57√398−72√570+3√906)+1144t2(27√94+2√398+36√570). |
Thus from (1.10),
GAM(Υ1)=∑rs∈E(Υ1)2√Mr×Ms(Mr+Ms). |
GAM(Υ1)=1|E1(Υ1(t))|+2√67|E2(Υ1(t))|+4√29|E3(Υ1(t))|+817|E4(Υ1(t))|+2√23|E5(Υ1(t))|+8√319|E6(Υ1(t))|+√32|E7(Υ1(t))|+2√67|E8(Υ1(t))|+1|E9(Υ1(t))|+2√67|E10(Υ1(t))|+1|E11(Υ1(t))|+4√37|E12(Υ1(t))|+2√23|E13(Υ1(t))|+4√611|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)+2√67(4t)+4√29(4)+817(4t−4)+2√23(4t−4)+8√319(4t−4)+√32(12t−8)+2√67(4t−4)+1(2t−2)+2√67(4t−4)+1(9t2−7t+3)+4√37(4)+2√23(4t−4)+4√611(4t−4)+√32(36t2−72t+36)+√12288152(4t−4)+1213(4t−4)+√12288112(4t−4)+√24576176(4t−4)+√18432136(4t−4)+2425(36t2−76t+40), |
⟹GAM(Υ1)=373771105−976√2153+202√319−400√677+(488√677+416√251−√33238133−3778295525)t+925(121+50√3)t2. |
Now, we are calculating certain degree-based multiplicative topological indices of the Υ2≅D2(t), where t∈N for second type of Dominating David Derived network.
Theorem 2.2.1. Consider the second type of Dominating David Derived network Υ2≅D2(t) for t∈N. 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(7t−4)(6+t(9t−14))(5+t(9t−13))(3+t(9t−11)). |
Proof. Let Υ2 be the second type of Dominating David Derived network. The D2(t) has 9t2+7t−5 vertices of degree 2, 18t2−26t+10 vertices of degree 3 and 27t2−33t+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)=∏r∈V(Υ2)(dr)2. |
(dr,ds) where rs∈E(Υ2) | Number of edges |
(2,2) | 4t |
(2,3) | 18t2−22t+6 |
(2,4) | 28t−16 |
(3,4) | 36t2−56t+24 |
(4,4) | 36t2−52t+20 |
By using vertex partitions, we get
II1(Υ2)=(2)2(9t2+7t−5)×(3)2(18t2−26t+10)×(4)2(27t2−33t+12),=4(9t2+7t−5)×9(18t2−26t+10)×16(27t2−33t+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)=∏rs∈E(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(18t2−22t+6)×8(28t−16)×12(36t2−56t+24)×16(36t2−52t+20), |
By doing some calculations, we have
⟹II2(Υ2)=18874368t(7t−4)(6+t(9t−14))(5+t(9t−13))(3+t(9t−11)). |
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 Υ2≅D2(t) be the second type of Dominating David Derived network, then
HII1(Υ2)=23121100800t(7t−4)(6+t(9t−14))(5+t(9t−13))(3+t(9t−11)), |
HII2(Υ2)=695784701952t(7t−4)(6+t(9t−14))(5+t(9t−13))(3+t(9t−11)). |
Proof. The outcome is obtained by using the edge partition provided in Table 3. The result is from (1.3),
HII1(Υ2)=∏rs∈E(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(18t2−22t+6)×36(28t−16)×49(36t2−56t+24)×64(36t2−52t+20), |
By doing some calculations, we get
⟹HII1(Υ2)=23121100800t(7t−4)(6+t(9t−14))(5+t(9t−13))(3+t(9t−11)). |
Also from (1.4),
HII2(Υ2)=∏rs∈E(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(18t2−22t+6)×64(28t−16)×144(36t2−56t+24)×256(36t2−52t+20), |
⟹HII2(Υ2)=695784701952t(7t−4)(6+t(9t−14))(5+t(9t−13))(3+t(9t−11)). |
Now, we calculate first and second Universal-Zagreb indices.
Theorem 2.2.3. Let Υ2≅D2(t) be the second type of Dominating David Derived network, then
MZa1(Υ2)=83+2a105at(7t−4)(6+t(9t−14))(5+t(9t−13))(3+t(9t−11)), |
MZa2(Υ2)=83+4a×9at(7t−4)(6+t(9t−14))(5+t(9t−13))(3+t(9t−11)). |
Proof. The outcome is obtained by using the edge partition provided in Table 3. The result is from (1.5),
MZa1(Υ2)=∏rs∈E(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(18t2−22t+6)×6a(28t−16)×7a(36t2−56t+24)×8a(36t2−52t+20), |
By doing some calculations, we get
⟹MZa1(Υ2)=83+2a105at(7t−4)(6+t(9t−14))(5+t(9t−13))(3+t(9t−11)). |
Also from (1.6),
MZa2(Υ2)=∏rs∈E(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(18t2−22t+6)×8a(28t−16)×12a(36t2−56t+24)×16a(36t2−52t+20), |
⟹MZa2(Υ2)=83+4a×9at(7t−4)(6+t(9t−14))(5+t(9t−13))(3+t(9t−11)). |
The sum and product connectivity of multiplicative indices are as follows.
Theorem 2.2.4. Let Υ2≅D2(t) be the second type of Dominating David Derived network, then
SCII(Υ2)=64√105t(7t−4)(9t2−14t+6)(9t2−13t+5)(9t2−11t+3), |
PCII(Υ2)=83t(7t−4)(9t2−14t+6)(9t2−13t+5)(9t2−11t+3). |
Proof. The outcome is obtained by using the edge partition provided in Table 3. The result is from (1.7),
SCII(Υ2)=∏rs∈E(Υ2)1√dr+ds. |
SCII(Υ2)=12|E1(Υ2(t))|×1√5|E2(Υ2(t))|×1√6|E3(Υ2(t))|×1√7|E4(Υ2(t))|×1√8|E5(Υ2(t))|,=12(4t)×1√5(18t2−22t+6)×1√6(28t−16)×1√7(36t2−56t+24)×1√8(36t2−52t+20), |
By doing some calculations, we get
⟹SCII(Υ2)=64√105t(7t−4)(9t2−14t+6)(9t2−13t+5)(9t2−11t+3). |
Also from (1.8),
PCII(Υ2)=∏rs∈E(Υ2)1√dr×ds. |
PCII(Υ2)=12|E1(Υ2(t))|×1√6|E2(Υ2(t))|×1√8|E3(Υ2(t))|×1√12|E4(Υ2(t))|×14|E5(Υ2(t))|,=12(4t)×1√6(18t2−22t+6)×1√8(28t−16)×1√12(36t2−56t+24)×1√16(36t2−52t+20), |
By doing some calculations, we get
⟹PCII(Υ2)=83t(7t−4)(9t2−14t+6)(9t2−13t+5)(9t2−11t+3). |
Now we calculate Multiple atom-bond connectivity index and Multiple Geometric-Arithmetic index.
Theorem 2.2.5. Let Υ2≅D2(t) be the second type of Dominating David Derived network, then
ABCM(Υ2)=14080(−1020√13−425√21−1360√23−510√29+1020√35−510√42−255√74+1700√78+3060√87−510√134−340√165−680√186−170√206−340√222+170√330−816√395+850√398−720√510−85√906)+14080t(6120+2040√13+2040√14+425√21+1360√23+510√29+510√42+255√74−4420√78−6120√87+510√134+340√165+1020√186+170√206+340√222+816√395−1615√398+720√510+85√906)+14080t2(3060√78+3060√87+765√398), |
GAM(Υ2)=(1468845525+1848296√269003−1550√3133−2176√6385)+t(−3157285525−4140967667√2+2120√3133+3232√6385)+t2(86425+13608√2451). |
Proof. The outcome is obtained by using the edge partition provided in Table 4. The result is from (1.9),
ABCM(Υ2)=∑rs∈E(Υ2)√Mr+Ms−2Mr×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(4t−4)+√23144(4t−4)+√1396(18t2−30t+14)+√1396(4t−4)+√37384(4t−4)+√31384(12t−8)+√55768(4t−4)+√791152(4t−4)+√13256(8t−4)+√21512(4t−4)+√29768(36t2−72t+36)+√551536(4)+√291024(4t−4)+√1516144(4t−4)+√1034608(4t−4)+√372048(4t−4)+√17512288(4t−4)+√151024(4t−4)+√19918432(36t2−76t+40), |
(Mr,Ms)whereuv∈E(Υ2) | Number of edges | (Mr,Ms)whereuv∈E(Υ2) | Number of edges |
(8, 8) | 4t | (32, 48) | 8t-4 |
(8, 48) | 4t | (32, 96) | 4t-4 |
(8, 64) | 4 | (32,144) | 36t2−72t+36 |
(8,128) | 4t-4 | (48, 64) | 4 |
(9, 16) | 4t-4 | (48,128) | 4t-4 |
(9, 32) | 18t2−30t+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) | 36t2−76t+40 |
By making some calculations, we are getting
⟹ABCM(Υ2)=14080(−1020√13−425√21−1360√23−510√29+1020√35−510√42−255√74+1700√78+3060√87−510√134−340√165−680√186−170√206−340√222+170√330−816√395+850√398−720√510−85√906)+14080t(6120+2040√13+2040√14+425√21+1360√23+510√29+510√42+255√74−4420√78−6120√87+510√134+340√165+1020√186+170√206+340√222+816√395−1615√398+720√510+85√906)+14080t2(3060√78+3060√87+765√398). |
Thus from (1.10),
GAM(Υ2)=∑rs∈E(Υ2)2√Mr×Ms(Mr+Ms). |
GAM(Υ2)=1|E1(Υ2(t))|+2√67|E2(Υ2(t))|+4√29|E3(Υ2(t))|+817|E4(Υ2(t))|+2425|E5(Υ2(t))|+24√241|E6(Υ2(t))|+4√37|E7(Υ2(t))|+8√319|E8(Υ2(t))|+√32|E9(Υ2(t))|+2√67|E10(Υ2(t))|+35|E11(Υ2(t))|+25√6|E12(Υ2(t))|+√32|E13(Υ2(t))|+6√211|E14(Υ2(t))|+4√37|E15(Υ2(t))|+4√611|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)+2√67(4t)+4√29(4)+817(4t−4)+2425(4t−4)+24√241(18t2−30t+14)+4√37(4t−4)+8√319(4t−4)+√32(12t−8)+2√67(4t−4)+35(4t−4)+2√65(8t−4)+√32(4t−4)+611√2(36t2−72t+36)+4√37(4)+4√611(4t−4)+√12288152(4t−4)+1213(4t−4)+√12288112(4t−4)+√24576176(4t−4)+√18432136(4t−4)+2425(36t2−76t+40), |
⟹GAM(Υ2)=(1468845525+1848296√269003−1550√3133−2176√6385)+t(−3157285525−414096√27667+2120√3133+3232√6385)+t2(86425+13608√2451). |
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 Υ3≅D3(t) for t∈N. The first and second multiplicative Zagreb indices are equal to
II1(Υ3)=384t(−1+3t)(22+t(−59+45t)), |
II2(Υ3)=32768t2(9t−5)(11+9t(2t−3)). |
Proof. Let Υ3 be the third type of Dominating David Derived network. The Υ3 has 18t2−6t vertices of degree 2 and 45t2−59t+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)=∏r∈V(Υ3)(dr)2. |
(dr,ds) where rs∈E(Υ3) | Number of edges |
(2,2) | 4t |
(2,4) | 36t2−20t |
(4,4) | 72t2−108t+44 |
By using vertex partition, we get
II1(Υ3)=(2)2(18t2−6t)×(4)2(45t2−59t+22),II1(Υ3)=4(18t2−6t)×16(45t2−59t+22), |
By making some calculations, we have
⟹II1(Υ3)=384t(−1+3t)(22+t(−59+45t)). |
Also from (1.2), we have
II2(Υ3)=∏rs∈E(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(36t2−20t)×16(72t2−108t+44), |
By making some calculations, we have
⟹II2(Υ3)=32768t2(9t−5)(11+9t(2t−3)). |
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 Υ3≅D3(t) be the third type of Dominating David Derived network, then
HII1(Υ3)=2359296t2(9t−5)(11+9t(2t−3)), |
HII2(Υ3)=16777216t2(9t−5)(11+9t(2t−3)). |
Proof. The outcome is obtained by using the edge partition provided in Table 5. The result is from (1.3)
HII1(Υ3)=∏pq∈E(Υ3)(dp+dq)2. |
HII1(Υ3)=16|E1(Υ2(t))|×36|E2(Υ2(t))|×64|E3(Υ2(t))|,=16(4t)×36(36t2−20t)×64(72t2−108t+44), |
By doing some calculations, we get
⟹HII1(Υ3)=2359296t2(9t−5)(11+9t(2t−3)). |
Also from (1.4),
HII2(Υ3)=∏pq∈E(Υ3)(dp×dq)2. |
HII2(Υ3)=16|E1(Υ3(t))|×64|E2(Υ3(t))|×256|E3(Υ3(t))|, |
HII2(Υ3)=16(4t)×64(36t2−20t)×256(72t2−108t+44), |
By doing some calculations, we get
⟹HII2(Υ3)=16777216t2(9t−5)(11+9t(2t−3)). |
Now the first and second Universal-Zagreb indices are defined as.
Theorem 2.3.3. Let Υ3≅D3(t) be the third type of Dominating David Derived network, then
MZa1(Υ3)=3a641+at2(9t−5)(11+9t(2t−3)), |
MZa2(Υ3)=82+3at2(9t−5)(11+9t(2t−3)). |
Proof. The outcome is obtained by using the edge partition provided in Table 5. The result is from (1.5),
MZa1(Υ3)=∏pq∈E(G)(dp+dq)a. |
MZa1(Υ3)=(4)a|E1(Υ3(t))|×(6)a|E2(Υ3(t))|×(8)a|E3(Υ3(t))|,=4a(4t)×6a(36t2−20t)×8a(72t2−108t+44), |
By doing some calculation, we get
⟹MZa1(Υ3)=3a641+at2(9t−5)(11+9t(2t−3)). |
Thus from (1.6),
MZa2(Υ3)=∏pq∈E(Υ3)(dp×dq)a. |
MZa2(Υ3)=(4)a|E1(Υ3(t))|×(8)a|E2(Υ3(t))|×(16)a|E3(Υ3(t))|,=4a(4t)×8a(36t2−20t)×16a(72t2−108t+44), |
⟹MZa2(Υ3)=82+3at2(9t−5)(11+9t(2t−3)). |
The sum and product connectivity of multiplicative indices are described as follows.
Theorem 2.3.4. Let Υ3≅D3(t) be the third type of Dominating David Derived network of type 3, then
SCII(Υ3)=8√3t2(9t−5)(11+9t(2t−3)), |
PCII(Υ3)=2√2t2(9t−5)(11+9t(2t−3)). |
Proof. The outcome is obtained by using the edge partition provided in Table 5. The result is from (1.7),
SCII(Υ3)=∏uv∈E(G)1√dp+dq. |
SCII(Υ3)=12|E1(Υ3(t))|×1√6|E2(Υ3(t))|×1√8|E3(Υ3(t))|,=12(4t)×1√6(36t2−20t)×1√8(72t2−108t+44), |
By doing some calculation, we get
⟹SCII(Υ3)=8√3t2(9t−5)(11+9t(2t−3)). |
Also from (1.8),
PCII(Υ3)=∏pq∈E(Υ3)1√dp×dq. |
PCII(Υ3)=12|E1(Υ3(t))|×1√8|E2(Υ3(t))|×14|E3(Υ3(t))|,=12(4t)×1√8(36t2−20t)×1√16(72t2−108t+44), |
By doing some calculations we get
⟹PCII(Υ3)=2√2t2(9t−5)(11+9t(2t−3)). |
The Multiple atom-bond connectivity index and Multiple Geometric-Arithmetic index are calculated as follows.
Theorem 2.3.5. Let Υ3≅D3(t) be the third type of Dominating David Derived network, then
ABCM(Υ3)=1192(−144√23+48√35−24√71+96√78−24√95−24√134−12√143−6√254+24√318+30√510+32√573)+1192t(168√14+144√23+48√35+24√71−264√78+24√95+24√134+12√143+6√254−90√318−57√510+32√573)+1192t2(216√78+54√318+27√510), |
GAM(Υ3)=(36t2−60t+36)+817(4t−4)+89√2(4t−4)+109√2(4t+4)+23√2(8t−8)+23√2(12t−12)+45(36t2−60t+16)+45(36t2−44t+16). |
Proof. The outcome is obtained by using the edge partition provided in Table 6. The result is from (1.9).
(Mr,Ms) where rs∈E(Υ3) | Number of edges | (Mr,Ms) where rs∈E(Υ3) | Number of edges |
(8, 8) | 4t | (64, 64) | 8t |
(8, 64) | 4t+4 | (64,128) | 8t-8 |
(8,128) | 4t-4 | (64,256) | 36t2−60t+16 |
(16, 32) | 12t-12 | (128,128) | 4t-4 |
(16, 64) | 36t2−44t+16 | (128,256) | 4t+4 |
(16,128) | 4t-4 | (256,256) | 36t2−76t+40 |
(32,256) | 4t-4 |
ABCM(Υ3)=∑rs∈E(Υ3)√Mr+Ms−2Mr×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(4t−4)+√23256(12t−12)+√39512(36t2−44t+16)+√711024(4t−4)+√1434096(4t−4)+√632048(8t)+√954096(8t−8)+√1598192(36t2−60t+16)+√1278192(4t−4)+√19116384(4t+4)+√25532768(36t2−76t+40), |
By doing some calculations, we get
⟹ABCM(Υ3)=1192(−144√23+48√35−24√71+96√78−24√95−24√134−12√143−6√254+24√318+30√510+32√573)+1192t(168√14+144√23+48√35+24√71−264√78+24√95+24√134+12√143+6√254−90√318−57√510+32√573)+1192t2(216√78+54√318+27√510). |
Also from (1.10),
GAM(Υ3)=∑rs∈E(Υ3)2√Mr×Ms(Mr+Ms). |
GAM(Υ3)=(1)|E1(Υ3(t))|+4√29|E2(Υ3(t))|+817|E3(Υ3(t))|+2√23|E4(Υ3(t))|+45|E5(Υ3(t))|+4√29|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)+4√29(4t+4)+817(4t−4)|+2√23(12t−12)+45(36t2−44t+16)+4√29(4t−4)+√8192144(4t−4)+(1)(8t)+√819296(8t−8)+45(36t2−60t+16)+(1)(4t−4)+√32768192(4t+4)+(1)(36t2−76t+40), |
⟹GAM(Υ3)=(36t2−60t+36)+817(4t−4)+89√2(4t−4)+109√2(4t+4)+23√2(8t−8)+23√2(12t−12)+45(36t2−60t+16)+45(36t2−44t+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.
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.
[1] |
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(dr,ds) where rs∈E(Υ1) | Number of edges |
(2,2) | 4t |
(2,3) | 4t−4 |
(2,4) | 28t−16 |
(3,3) | 9t2−13t+5 |
(3,4) | 36t2−56t+24 |
(4,4) | 36t2−52t+20 |
(Mr,Ms) where rs∈E(Υ1) | Number of edges | (Mr,Ms) where rs∈E(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) | 36t2−72t+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) | 36t2−76t+40 |
(48, 48) | 9t2−7t+3 |
(dr,ds) where rs∈E(Υ2) | Number of edges |
(2,2) | 4t |
(2,3) | 18t2−22t+6 |
(2,4) | 28t−16 |
(3,4) | 36t2−56t+24 |
(4,4) | 36t2−52t+20 |
(Mr,Ms)whereuv∈E(Υ2) | Number of edges | (Mr,Ms)whereuv∈E(Υ2) | Number of edges |
(8, 8) | 4t | (32, 48) | 8t-4 |
(8, 48) | 4t | (32, 96) | 4t-4 |
(8, 64) | 4 | (32,144) | 36t2−72t+36 |
(8,128) | 4t-4 | (48, 64) | 4 |
(9, 16) | 4t-4 | (48,128) | 4t-4 |
(9, 32) | 18t2−30t+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) | 36t2−76t+40 |
(dr,ds) where rs∈E(Υ3) | Number of edges |
(2,2) | 4t |
(2,4) | 36t2−20t |
(4,4) | 72t2−108t+44 |
(Mr,Ms) where rs∈E(Υ3) | Number of edges | (Mr,Ms) where rs∈E(Υ3) | Number of edges |
(8, 8) | 4t | (64, 64) | 8t |
(8, 64) | 4t+4 | (64,128) | 8t-8 |
(8,128) | 4t-4 | (64,256) | 36t2−60t+16 |
(16, 32) | 12t-12 | (128,128) | 4t-4 |
(16, 64) | 36t2−44t+16 | (128,256) | 4t+4 |
(16,128) | 4t-4 | (256,256) | 36t2−76t+40 |
(32,256) | 4t-4 |
(dr,ds) where rs∈E(Υ1) | Number of edges |
(2,2) | 4t |
(2,3) | 4t−4 |
(2,4) | 28t−16 |
(3,3) | 9t2−13t+5 |
(3,4) | 36t2−56t+24 |
(4,4) | 36t2−52t+20 |
(Mr,Ms) where rs∈E(Υ1) | Number of edges | (Mr,Ms) where rs∈E(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) | 36t2−72t+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) | 36t2−76t+40 |
(48, 48) | 9t2−7t+3 |
(dr,ds) where rs∈E(Υ2) | Number of edges |
(2,2) | 4t |
(2,3) | 18t2−22t+6 |
(2,4) | 28t−16 |
(3,4) | 36t2−56t+24 |
(4,4) | 36t2−52t+20 |
(Mr,Ms)whereuv∈E(Υ2) | Number of edges | (Mr,Ms)whereuv∈E(Υ2) | Number of edges |
(8, 8) | 4t | (32, 48) | 8t-4 |
(8, 48) | 4t | (32, 96) | 4t-4 |
(8, 64) | 4 | (32,144) | 36t2−72t+36 |
(8,128) | 4t-4 | (48, 64) | 4 |
(9, 16) | 4t-4 | (48,128) | 4t-4 |
(9, 32) | 18t2−30t+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) | 36t2−76t+40 |
(dr,ds) where rs∈E(Υ3) | Number of edges |
(2,2) | 4t |
(2,4) | 36t2−20t |
(4,4) | 72t2−108t+44 |
(Mr,Ms) where rs∈E(Υ3) | Number of edges | (Mr,Ms) where rs∈E(Υ3) | Number of edges |
(8, 8) | 4t | (64, 64) | 8t |
(8, 64) | 4t+4 | (64,128) | 8t-8 |
(8,128) | 4t-4 | (64,256) | 36t2−60t+16 |
(16, 32) | 12t-12 | (128,128) | 4t-4 |
(16, 64) | 36t2−44t+16 | (128,256) | 4t+4 |
(16,128) | 4t-4 | (256,256) | 36t2−76t+40 |
(32,256) | 4t-4 |