# On Sum Degree-Based Topological Indices of Some Connected Graphs

Authors: Muhammad Tanveer Hussain; Navid Iqbal; Usman Babar; Tabasam Rashid; Md Nur Alam
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In mathematical chemistry, molecular structure of any chemical substance can be expressed by a numeric number or polynomial or sequence of number which represents the whole graph is called topological index. An important branch of graph theory is the chemical graph theory. As a consequence of their worldwide uses, chemical networks have inspired researchers since their development. Determination of the expressions for topological indices of different derived graphs of graphs is a new and interesting problem in graph theory. In this article, some graphs which are derived from honeycomb structure are studied and obtained their exact results for sum degree-based indices. Additionally, a comparison is shown graphically among all the indices.


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## Body

## 1. Introduction and Preliminary Results

Atoms are connected through covalent bonds in molecular frames. Atoms are called vertices and covalent bonds as edges in graph theory. Information sciences, Mathematics, and Chemistry are combined in Cheminformatics. This is a new research field that attracts the attention of researchers. Dominating David Derived networks are under consideration in the present research article which are further extracted from honeycomb structures. Honeycomb structures taken from bee honeycombs had observed a lot of applications in different fields including chemical engineering, transportation, mechanical engineering, architecture, nanofabrication, and recently bioinformatics. The greatest challenge in this research direction is to comprehend the unique characteristics of honeycomb structures, relying on their structures, scales, and the materials used to allow the minimization of the materials to get minimum weight and maximum strength. Nowadays, there is a substantial use of honeycomb networks in computer graphics, processing of images, chemistry as the portrayal of benzenoid hydrocarbons, and cellular phone base stations by using hexagon arrangements with repetitive building of honeycomb structures [1].A topological indexTI, sometimes also known as a graph-theoretic index, is a numerical invariant of a chemical graph [2]. There are many types of TI′s, but most popular and authentic TI′s are distance-based, degree-based, and neighbourhood degree-based indices. These indices contain a lot of information within themselves.The method of drawing Dominating David Derived networks (dimensiont) is as follows:Step 1: consider a honeycomb networkHCt dimension t as shown in Figure 1(a).Step 2: split each edge into two by embedding another vertex as shown in Figure1(b).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 as shown in Figure1(c).Step 4: place vertices at new edge crossings as shown in Figure1(d).Step 5: remove initial vertices and edges of honeycomb network as shown in Figure1(e).Step 6: split each horizontal edge into two edges by inserting a new vertex. The resulting graph is called Dominating David Derived systemDDDt of measurement t as shown in Figure 1(f) [1, 3].Figure 1
Construction algorithm for Dominating David Derived networkDDD2. (a) Step 1. (b) Step 2. (c) Step 3. (d) Step 4. (e) Step 5. (f) Step 6 [3].
(a)(b)(c)(d)(e)(f)The first type of Dominating David Derived networkD1t can be obtained by connecting vertices of degree two by an edge, which are not in the boundary (see Figure 2).Figure 2
First type of Dominating David Derived networkD12 [3].The second type of Dominating David Derived networkD2t can be obtained by subdividing once the new edge introduced in D1t (see Figure 3).Figure 3
Second type of Dominating David Derived networkD22 [3].The third type of Dominating David Derived networkD3t can be obtained from D1t by introducing parallel path of length 2 between the vertices of degree two which are not in the boundary. See Figure 4 for third type of Dominating David Derived network of dimension 2, D32. Moreover, isomorphic graph of Dominating David Derived network of dimension 2 can be seen in Figure 5.Figure 4
Third type of Dominating David Derived networkD32 [3].Figure 5
Isomorphic graph of DDD(2) [3].In this article,Υ is considered a network with a VΥ vertex set and an edge set of EΥ, and dr is the degree of vertex r∈VΥ. Let SΥr denote the sum of the degrees of all vertices adjacent to a vertex r. Graovac et al. [4] defined fifth M-Zagreb indices as polynomials for a molecular graph and these are characterized as follows.LetΥ be a graph. Then,(1)M1G5ϒ=∑rs∈EΥSGr+SGs,(2)M2G5ϒ=∑rs∈EΥSGr×SGs.Kulli [5, 6] motivated by above indices described some new topological indices and named them as the fifth M-Zagreb indices of first and second type and fifth hyper-M-Zagreb indices of first and second type of a graph ϒ. They are defined as(3)M1aG5ϒ=∑rs∈EΥSGr×SGsa,(4)M2aG5ϒ=∑rs∈EΥSGr×SGsa,(5)HM12G5ϒ=∑rs∈EΥSGr×SGs2,(6)HM22G5ϒ=∑rs∈EΥSGr×SGs2,They also define a new version of Zagreb index which they call as third Zagreb index or fifthM3-Zagreb [7].(7)M3G5ϒ=∑rs∈EΥSGr−SGs.


## 2. Results

We have studied the new topological indices described by Kulli named as fifth M-Zagreb indices, fifth M-Zagreb polynomials, andM3-Zagreb index and give closed formulae of these indices for Dominating David Derived networks. In the following publications, the authors found out and described many other interesting characterizations for topological indices for different networks. For example, Ali et al. [8] studied degree-based topological indices and computed some new topological indices [9] for various networks. Bača et al. [10] and Baig et al. [11] found some topological indices of carbon nanotube network and polyoxide networks, respectively. Baig et al. [12] and Imran [13] studied about topological polynomials of certain nanostructures and mesh-derived networks, respectively. Eliasi et al. [14] and Liu et al. [15] computed multiplicative indices of first Zagreb and carbon graphite t-levels, respectively. Furthermore, Gao et al. [16] computed multiple ABC and GA index; Imran et al. [17] studied topological properties of Dominating David Derived network; Imran et al. [18] investigated topological properties of diamond-like networks. To know more about topological indices of various graph families, see [19, 20]. For the basic notations and definitions, see [21, 22].


### 2.1. Results for First Type of Dominating David Derived Networks

In this section, we calculate degree-based topological indices of the dimensiont for first type of Dominating David Derived networks. In the coming theorems, we compute M-Zagreb indices and polynomials.Theorem 1.
Letϒ1≅D1t be the first type of Dominating David Derived network, then the first and second fifth M-Zagreb indices are equal to(8)M1G5ϒ1=1002−2714t+2178t2,M2G5ϒ1=8065−20581t+14697t2.(9)M1G5ϒ=∑rs∈EΥSGr+SGs.M1G5ϒ1=6+6E1Υ1t+6+11E2Υ1t+6+12E3Υ1t+6+14E4Υ1t+7+9E5Υ1t+7+12E6Υ1t+8+11E7Υ1t+8+13E8Υ1t+9+9E9ϒ1t+9+14E10ϒ1t+11+11E11ϒ1t+11+12E12ϒ1t+11+13E13ϒ1t+11+14E14ϒ1t+11+16E15ϒ1t+12+14E16ϒ1t+13+14E17ϒ1t+13+16E18ϒ1t+14+14E19ϒ1t+14+16E20ϒ1t,=6+64t+6+114t+6+124+6+144t−4+7+94t−4+7+124t−4+8+1112t−8+8+134t−4+9+92t−2+9+144t−4+11+119t2−7t+3+11+124+11+134t−4+11+1436t2−68t+32+11+164t−4+12+144t−4+13+144t−4+13+164t−4+14+144t−4+14+1636t2−76t+40.Proof.
The outcome can be obtained by using the edge partition in Table1. By using equation (1),
By doing some calculations, we get(10)⇒M1G5ϒ1=1002−2714t+2178t2.
Thus, from equation (2),(11)M2G5ϒ=∑rs∈EΥSGr+SGs,M2G5ϒ1=6×6E1ϒ1t+6×11E2ϒ1t+6×12E3ϒ1t+6×14E4ϒ1t+7×9E5ϒ1t+7×12E6ϒ1t+8×11E7ϒ1t+8×13E8ϒ1t+9×9E9ϒ1t+9×14E10ϒ1t+11×11E11ϒ1t+11×12E12ϒ1t+11×13E13ϒ1t+11×14E14ϒ1t+11×16E15ϒ1t+12×14E16ϒ1t+13×14E17ϒ1t+13×16E18ϒ1t+14×14E19ϒ1t+14×16E20ϒ1t,=6×64t+6×114t+6×124+6×144t−4+7×94t−4+7×124t−4+8×1112t−8+8×134t−4+9×92t−2+9×144t−4+11×119t2−7t+3+11×124+11×134t−4+11×1436t2−68t+32+11×164t−4+12×144t−4+13×144t−4+13×164t−4+14×144t−4+14×1636t2−76t+40,
By doing some calculations, we get(12)⇒M2G5ϒ1=8065−20581t+14697t2.Table 1
Edge partition of first type of Dominating David Derived networkD1t based on sum of degrees of end vertices of each edge.
Sr,Ss where rs∈Eϒ1Number of edgesSr,Ss where rs∈Eϒ1Number of edges6,64t11,119t2−7t+36,114t11,1246,12411,134t−46,144t−411,1436t2−68t+327,94t−411,164t−47,124t−412,144t−48,1112t−813,144t−48,134t−413,164t−49,92t−214,144t−49,144t−414,1636t2−76t+40Theorem 2.
Consider the first type of DDD networkΥ1≅D1t for t∈ℕ. Then the first and second general fifth M-Zagreb indices are equal to(13)M1aG5ϒ1=−22+3a3a−41+2a−41+a5a+23+a3a51+a−41+a7a−21+a9a+22+a9a−22+a13a−12×19a−4×21a+3×22a+32×25a−8×27a−4×29a+t22+3a3a+3a41+a+41+2a+41+a5a+41+a7a+21+a9a+22+a13a−19×22+a×15a+4×17a+16×19a+4×21a−7×22a+4×23a−68×25a+8×27a+4×29a+t25a62+a+9×22a+36×25a,M2aG5ϒ1=5×23+5a×7a+22+3a9a−41+2a11a−81+a11a−22+3a13a−41+2a13a−23+2a21a−21a22+3a+41+a33a−41+a49a−4×63a−22+a63a+25+a77a−2×81a−22+a91a+121a×3−4×143a+t−19×22+5a×7a+41+a9a+3×22+3a×11a+41+2a11a+22+3a×13a+41+2a13a+23+2a21a+22+3a21a+22+a33a+41+a49a+4×63a+22+a63a−17×22+a×77a+2×81a+22+a91a−7×121a+4×143a+t29×22+5a×7a+9×22+a×77a+9×121a.Proof.
LetΥ1 be the first type of DDD network. Table 1 shows such an edge partition of D1t. Thus, from equation (3), it follows that(14)M1aG5ϒ=∑rs∈EΥSGr+SGsa,
By using edge partitions in Table1, we get(15)M1G5ϒ1=6+6E1Υ1t+6+11E2Υ1t+6+12E3Υ1t+6+14E4Υ1t+7+9E5Υ1t+7+12E6Υ1t+8+11E7Υ1t+8+13E8Υ1t+9+9E9ϒ1t+9+14E10ϒ1t+11+11E11ϒ1t+11+12E12ϒ1t+11+13E13ϒ1t+11+14E14ϒ1t+11+16E15ϒ1t+12+14E16ϒ1t+13+14E17ϒ1t+13+16E18ϒ1t+14+14E19ϒ1t+14+16E20ϒ1t,=6+64t+6+114t+6+124+6+144t−4+7+94t−4+7+124t−4+8+1112t−8+8+134t−4+9+92t−2+9+144t−4+11+119t2−7t+3+11+124+11+134t−4+11+1436t2−68t+32+11+164t−4+12+144t−4+13+144t−4+13+164t−4+14+144t−4+14+1636t2−76t+40,
By doing some calculations, we have(16)⇒M1aG5ϒ1=−22+3a3a−41+2a−41+a5a+23+a3a51+a−41+a7a−21+a9a+22+a9a−22+a13a−12×19a−4×21a+3×22a+32×25a−8×27a−4×29a+t22+3a3a+3a41+a+41+2a+41+a5a+41+a7a+21+a9a+22+a13a−19×22+a×15a+4×17a+16×19a+4×21a−7×22a+4×23a−68×25a+8×27a+4×29a+t25a62+a+9×22a+36×25a,
From equation (4), we have(17)M2aG5ϒ=∑rs∈EΥSGr+SGsa,
By using edge partitions in Table1, we get(18)M2aG5ϒ1=6×62E1ϒ1t+6×112E2ϒ1t+6×122E3ϒ1t+6×142E4ϒ1t+7×92E5ϒ1t+7×122E6ϒ1t+8×112E7ϒ1t+8×132E8ϒ1t+9×92E9ϒ1t+9×142E10ϒ1t+11×112E11ϒ1t+11×122E12ϒ1t+11×132E13ϒ1t+11×142E14ϒ1t+11×162E15ϒ1t+12×142E16ϒ1t+13×142E17ϒ1t+13×162E18ϒ1t+14×142E19ϒ1t+14×162E20ϒ1t,=6×624t+6×1124t+6×1224+6×1424t−4+7×924t−4+7×1224t−4+8×11212t−8+8×1324t−4+9×922t−2+9×1424t−4+11×1129t2−7t+3+11×1224+11×1324t−4+11×14236t2−68t+32+11×1624t−4+12×1424t−4+13×1424t−4+13×1624t−4+14×1424t−4+14×16236t2−76t+40.
By doing some calculations, we have(19)⇒M2aG5ϒ1=5×23+5a×7a+22+3a9a−41+2a11a−81+a11a−22+3a13a−41+2a13a−23+2a21a−21a22+3a+41+a33a−41+a49a−4×63a−22+a63a+25+a77a−2×81a−22+a×91a+121a×3−4×143a+t−19×22+5a×7a+41+a9a+3×22+3a×11a+11a41+2a+22+3a×13a+41+2a13a+23+2a21a+22+3a21a+22+a33a+41+a49a+63a×4+22+a63a−17×22+a×77a+2×81a+22+a91a−7×121a+4×143a+t29×22+5a×7a+9×22+a×77a+9×121a.Theorem 3.
Consider the first type of DDD networkϒ1≅D1t for t∈ℕ. Then, the first and second hyper fifth M-Zagreb indices are equal to(20)HM12G5ϒ1=84005−10286t+7407t2,HM22G5ϒ1=1868329−4442989t+2791881t2.Proof.
Letϒ1 be the first type of DDD network. Table 1 shows such an edge partition of D1t. Thus, from equation (5), it follows that(21)HM12G5ϒ=∑rs∈EΥSGr+SGs2.
By using edge partitions in Table1, we get(22)HM12G5ϒ1=6+62E1Υ1t+6+112E2Υ1t+6+122E3Υ1t+6+142E4Υ1t+7+92E5Υ1t+7+122E6Υ1t+8+112E7Υ1t+8+132E8Υ1t+9+92E9ϒ1t+9+142E10ϒ1t+11+112E11ϒ1t+11+122E12ϒ1t+11+132E13ϒ1t+11+142E14ϒ1t+11+162E15ϒ1t+12+142E16ϒ1t+13+142E17ϒ1t+13+162E18ϒ1t+14+142E19ϒ1t+14+162E20ϒ1t,=6+64t+6+114t+6+124+6+144t−4+7+94t−4+7+124t−4+8+11212t−8+8+1324t−4+9+922t−2+9+1424t−4+11+1129t2−7t+3+11+1224+11+1324t−4+11+14236t2−68t+32+11+1624t−4+12+1424t−4+13+1424t−4+13+1624t−4+14+1424t−4+14+16236t2−76t+40.
By doing some calculations, we have(23)⇒HM12G5ϒ1=84005−10286t+7407t2.
From equation (6), we have(24)HM22G5ϒ=∑rs∈EΥSGr×SGs2.
By using edge partitions in Table1, we get(25)HM22G5ϒ1=6×62E1ϒ1t+6×112E2ϒ1t+6×122E3ϒ1t+6×142E4ϒ1t+7×92E5ϒ1t+7×122E6ϒ1t+8×112E7ϒ1t+8×132E8ϒ1t+9×92E9ϒ1t+9×142E10ϒ1t+11×112E11ϒ1t+11×122E12ϒ1t+11×132E13ϒ1t+11×142E14ϒ1t+11×162E15ϒ1t+12×142E16ϒ1t+13×142E17ϒ1t+13×162E18ϒ1t+14×142E19ϒ1t+14×162E20ϒ1t,=6×624t+6×1124t+6×1224+6×1424t−4+7×924t−4+7×1224t−4+8×11212t−8+8×1324t−4+9×922t−2+9×1424t−4+11×1129t2−7t+3+11×1224+11×1324t−4+11×14236t2−68t+32+11×1624t−4+12×1424t−4+13×1424t−4+13×1624t−4+14×1424t−4+14×16236t2−76t+40.
By doing some calculations, we have(26)⇒HM22G5ϒ1=1868329−4442989t+2791881t2.Theorem 4.
Consider the first type of DDD networkϒ1≅D1t for t∈ℕ. Then, the third M-Zagreb index is equal to(27)M3G5ϒ1=47−37t+45t2.Proof.
Letϒ1 be the first type of DDD network. Table 1 shows such an edge partition of D1t. Thus, from equation (7), it follows that(28)M3G5ϒ=∑rs∈EΥSGr×SGs.
By using edge partitions in Table1, we get(29)M3G5ϒ1=6−6E1ϒ1t+6−11E2ϒ1t+6−12E3ϒ1t+6−14E4ϒ1t+7−9E5ϒ1t+7−12E6ϒ1t+8−11E7ϒ1t+8−13E8ϒ1t+9−9E9ϒ1t+9−14E10ϒ1t+11−11E11ϒ1t+11−12E12ϒ1t+11−13E13ϒ1t+11−14E14ϒ1t+11−16E15ϒ1t+12−14E16ϒ1t+13−14E17ϒ1t+13−16E18ϒ1t+14−14E19ϒ1t+14−16E20ϒ1t+,=6−64t+6−114t+6−124+6−144t−4+7−94t−4+7−124t−4+8−1112t−8+8−134t−4+9−92t−2+9−144t−4+11−119t2−7t+3+11−124+11−134t−4+11−1436t2−68t+32+11−164t−4+12−144t−4+13−144t−4+13−164t−4+14−144t−4+14−1636t2−76t+40.
By doing some calculations, we have(30)⇒M3G5ϒ1=47−37t+45t2.


### 2.2. Results for Second Type of Dominating David Derived Network

Now, we are calculating fifth M-Zagreb topological indices of theϒ2≅D2t, where t∈ℕ for second type of Dominating David Derived network.Theorem 5.
Letϒ2≅D2t be the second type of Dominating David Derived network, then the first and second fifth M-Zagreb indices are equal to(31)M1G5ϒ2=8129−349t+279t2,M2G5ϒ2=24321−823t+591t2.Proof.
The outcome can be obtained by using the edge partition in Table2. By using equation (1),
By doing some calculations, we get(32)⇒M1G5ϒ2=8129−349t+279t2.
Thus, from equation (2),(33)M2G5ϒ=∑rs∈EΥSGr×SGs,M2G5ϒ2=6×6E1ϒ2t+6×8E2ϒ2t+6×10E3ϒ2t+6×11E4ϒ2t+6×12E5ϒ2t+6×14E6ϒ2t+7×8E7ϒ2t+7×12E8ϒ2t+8×11E9ϒ2t+8×13E10ϒ2t+8×14E11ϒ2t+10×11E12ϒ2t+10×13E13ϒ2t+10×14E14ϒ2t+11×12E15ϒ2t+11×14E16ϒ2t+11×16E17ϒ2t+12×14E18ϒ2t+13×14E19ϒ2t+13×16E20ϒ2t+14×14E21ϒ2t+14×16E22ϒ2t,=6×64t+6×84t−4+6×1018t2−30t+14+6×114t+6×124+6×144t−4+7×84t−4+7×124t−4+8×1112t−8+8×134t−4+4t−4+10×118t−4+10×134t−4+10×1436t2−72t+36+11×124+11×144t−4+11×164t−4+12×144t−4+13×144t−4+13×164t−4+14×144t−4+14×1636t2−76t+40.
By doing some calculations, we get(34)⇒M2G5ϒ2=24321−823t+591t2.Table 2
Edge partition of second type of Dominating David Derived networkD2t based on sum of degrees of end vertices of each edge.
Sr,Ss where rs∈Eϒ2Number of edgesSr,Ss where rs∈Eϒ2Number of edges6,64t10,118t−46,84t−410,134t−46,1018t2−30n+1410,1436t2−72t+366,114t11,1246,12411,144t−46,144t−411,164t−47,84t−412,144t−47,124t−413,144t−48,1112t−813,164t−48,134t−414,144t−48,144t−414,1636t2−76t+40(35)M1G5ϒ=∑rs∈EΥSGr×SGs,M1G5ϒ2=6+6E1ϒ2t+6+11E2ϒ2t+6+12E3ϒ2t+6+14E4ϒ2t+7+9E5ϒ2t+7+12E6ϒ2t+8+11E7ϒ2t+8+13E8ϒ2t+9+9E9ϒ2t+9+14E10ϒ2t+11+11E11ϒ2t+11+12E12ϒ2t+11+13E13ϒ2t+11+14E14ϒ2t+11+16E15ϒ2t+12+14E16ϒ2t+13+14E17ϒ2t+13+16E18ϒ2t+14+14E19ϒ2t+14+16E20ϒ2t,=6+64t+6+114t+6+124+6+144t−4+7+94t−4+7+124t−4+8+1112t−8+8+134t−4+9+92t−2+9+144t−4+11+119t2−7t+3+11+124+11+134t−4+11+1436t2−68t+32+11+164t−4+12+144t−4+13+144t−4+13+164t−4+14+144t−4+14+1636t2−76t+40.Theorem 6.
Consider the second type of DDD networkϒ2≅D2t for t∈ℕ. Then the first and second general fifth M-Zagreb indices are equal to(36)M1aG5ϒ2=7×21+4a+22+3a32+a−41+a5a+23+a3a51+a−22+a7a−41+a7a+22+a9a−22+a11a−22+a13a−4×15a−12×19a−8×21a−4×25a−8×27a−4×29a+t−15×21+4a−23+3a32+a+3a41+a+41+a5a+22+a7a+4×31+a7a+41+a7a+22+a11a+22+a13a+4×15a−19×22+a15a+4×17a+16×19a+4×23a+4×25a+8×27a+4×29a+t29×21+4a+22+3a32+a+5a62+a,M2aG5ϒ2=22+3a9a+41+a33a+23+2a21at−1+9×41+a35at−12+t41+a9a+t22+a33a+48a4t−4+56a4t−4+104a4t−4+112a4t−4+130a4t−4+154a4t−4+168a4t−4+176a4t−4+182a4t−4+196a4t−4+208a4t−4+110a8t−4+88a12t−8+60a18t2−30t+14+224a36t2−76t+40.Proof.
Letϒ2 be the second type of DDD network. Table 2 shows such an edge partition of D2t. Thus, from equation (3), it follows that(37)M1aG5ϒ=∑rs∈EΥSGr+SGsa.
By using edge partitions in Table2, we get(38)M1aG5ϒ2=6+6aE1ϒ2t+6+8aE2ϒ2t+6+10aE3ϒ2t+6+11aE4ϒ2t+6+12aE5ϒ2t+6+14aE6ϒ2t+7+8aE7ϒ2t+7+12aE8ϒ2t+8+11aE9ϒ2t+8+13aE10ϒ2t+8+14aE11ϒ2t+10+11aE12ϒ2t+10+13aE13ϒ2t+10+14aE14ϒ2t+11+12aE15ϒ2t+11+14aE16ϒ2t+11+16aE17ϒ2t+12+14aE18ϒ2t+13+14aE19ϒ2t+13+16aE20ϒ2t+14+14aE21ϒ2t+14+16aE22ϒ2t,=6+6a4t+6+8a4t−4+6+10a18t2−30t+14+6+11a4t+6+12a4+6+14a4t−4+7+8a4t−4+7+12a4t−4+8+11a12t−8+8+13a4t−4+8+14a4t−4+10+11a8t−4+10+13a4t−4+10+14a36t2−72t+36+11+12a4+11+14a4t−4+11+16a4t−4+12+14a4t−4+13+14a4t−4+13+16a4t−4+14+14a4t−4+14+16a36t2−76t+40.
By doing some calculations, we have(39)⇒M1aG5ϒ2=7×21+4a+22+3a32+a−41+a5a+23+a3a51+a−22+a7a−41+a7a+22+a9a−22+a11a−22+a13a−4×15a−12×19a−8×21a−4×25a−8×27a−4×29a+t−15×21+4a−23+3a32+a+3a41+a+41+a5a+22+a7a+4×31+a7a+41+a7a+22+a11a+13a22+a+4×15a−19×22+a15a+4×17a+16×19a+4×23a+4×25a+8×27a+4×29a+t29×21+4a+22+3a32+a+5a62+a.
From equation (4), we have(40)M2aG5ϒ=∑rs∈EΥSGr+SGsa.
By using edge partitions in Table2, we get(41)M2G5ϒ2=6×6aE1ϒ2t+6×8aE2ϒ2t+6×10aE3ϒ2t+6×11aE4ϒ2t+6×12aE5ϒ2t+6×14aE6ϒ2t+7×8aE7ϒ2t+7×12aE8ϒ2t+8×11aE9ϒ2t+8×13aE10ϒ2t+8×14aE11ϒ2t+10×11aE12ϒ2t+10×13aE13ϒ2t+10×14aE14ϒ2t+11×12aE15ϒ2t+11×14aE16ϒ2t+11×16aE17ϒ2t+12×14aE18ϒ2t+13×14aE19ϒ2t+13×16aE20ϒ2t+14×14aE21ϒ2t+14×16aE22ϒ2t,=6×6a4t+6×8a4t−4+6×10a18t2−30t+14+6×11a4t+6×12a4+6×14a4t−4+7×8a4t−4+7×12a4t−4+8×11a12t−8+8×13a4t−4+4t−4+10×11a8t−4+10×13a4t−4+10×14a36t2−72t+36+11×12a4+11×14a4t−4+11×16a4t−4+12×14a4t−4+13×14a4t−4+13×16a4t−4+14×14a4t−4+14×16a36t2−76t+40.
By doing some calculations, we have(42)⇒M2aG5ϒ2=22+3a9a+41+a33a+23+2a21at−1+9×41+a35at−12+t41+a9a+t22+a33a+48a4t−4+56a4t−4+104a4t−4+112a4t−4+130a4t−4+154a4t−4+168a4t−4+176a4t−4+182a4t−4+196a4t−4+208a+4t−4110a8t−4+88a12t−8+60a18t2−30t+14+224a36t2−76t+40.Theorem 7.
Consider the second type of DDD networkϒ2≅D2t for t∈ℕ. Then the first and second hyper fifth M-Zagreb indices are equal to(43)HM12G5ϒ2=83875−9985t+7218t2,HM22G5ϒ2=3253531−127816t+80523t2.Proof.
Letϒ2 be the second type of DDD network. Table 2 shows such an edge partition of D2t. Thus, from equation (5), it follows that(44)HM12G5ϒ=∑rs∈EΥSGr+SGs2.
By using edge partitions in Table2, we get(45)M12G5ϒ2=6+62E1ϒ2t+6+82E2ϒ2t+6+102E3ϒ2t+6+112E4ϒ2t+6+122E5ϒ2t+6+142E6ϒ2t+7+82E7ϒ2t+7+122E8ϒ2t+8+112E9ϒ2t+8+132E10ϒ2t+8+142E11ϒ2t+10+112E12ϒ2t+10+132E13ϒ2t+10+142E14ϒ2t+11+122E15ϒ2t+11+142E16ϒ2t+11+162E17ϒ2t+12+142E18ϒ2t+13+142E19ϒ2t+13+162E20ϒ2t+14+142E21ϒ2t+14+162E22ϒ2t,=6+624t+6+824t−4+6+10218t2−30t+14+6+1124t+6+1224+6+1424t−4+7+824t−4+7+1224t−4+8+11212t−8+8+1324t−4+8+1424t−4+10+1128t−4+10+1324t−4+10+14236t2−72t+36+11+1224+11+1424t−4+11+1624t−4+12+1424t−4+13+1424t−4+13+1624t−4+14+1424t−4+14+16236t2−76t+40.
By doing some calculations, we have(46)⇒HM12G5ϒ2=83875−9985t+7218t2.
From equation (6), we have(47)HM22G5ϒ=∑rs∈EΥSGr+SGs2.
By using edge partitions in Table2, we get(48)M22G5ϒ2=6×62E1ϒ2t+6×82E2ϒ2t+6×102E3ϒ2t+6×112E4ϒ2t+6×122E5ϒ2t+6×142E6ϒ2t+7×82E7ϒ2t+7×122E8ϒ2t+8×112E9ϒ2t+8×132E10ϒ2t+8×142E11ϒ2t+10×112E12ϒ2t+10×132E13ϒ2t+10×142E14ϒ2t+11×122E15ϒ2t+11×142E16ϒ2t+11×162E17ϒ2t+12×142E18ϒ2t+13×14aE19ϒ2t+13×162E20ϒ2t+14×142E21ϒ2t+14×162E22ϒ2t,=6×624t+6×824t−4+6×10a18t2−30t+14+6×1124t+6×1224+6×1424t−4+7×824t−4+7×1224t−4+8×11212t−8+8×1324t−4+4t−4+10×1128t−4+10×1324t−4+10×14236t2−72t+36+11×1224+11×1424t−4+11×1624t−4+12×1424t−4+13×1424t−4+13×1624t−4+14×1424t−4+14×16236t2−76t+40.
By doing some calculations, we have(49)⇒HM22G5ϒ2=3253531−127816t+80523t2.Theorem 8.
Consider the second type of DDD networkΥ2≅D2t for t∈ℕ. Then the third M-Zagreb index is equal to(50)M3G5ϒ2=813−40t+36t2.Proof.
LetΥ2 be the second type of DDD network. Table 2 shows such an edge partition of D2t. Thus, from equation (7), it follows that(51)M3G5ϒ=∑rs∈EΥSGr+SGs.
By using edge partitions in Table2, we get(52)M3G5Υ2=|6−6E1Υ2t+6−8E2Υ2t+6−10E3Υ2t+6−11E4Υ2t+6−12E5Υ2t+6−14E6Υ2t+7−8E7Υ2t+7−12E8Υ2t+8−11E9Υ2t+8−13E10Υ2t+8−14E11Υ2t+10−11E12Υ2t+10−13E13Υ2t+10−14E14Υ2t+11−12E15Υ2t+11−14E16Υ2t+11−16E17Υ2t+12−14E18Υ2t+13−14E19Υ2t+13−16E20Υ2t+14−14E21Υ2t+14−16E22Υ2t+,=6−64t+|6−84t−4+6−1018t2−30t+14+6−114t+6−124+6−144t−4+7−84t−4+7−124t−4+8−1112t−8+8−134t−4+8−144t−4+|10−11|8t−4+10−134t−4+10−1436t2−72t+36+411−12+11−144t−4+11−164t−4+12−144t−4+13−144t−4+13−164t−4+14−144t−4+|14−16|36t2−76t+40.
By doing some calculations, we have(53)⇒M3G5ϒ2=813−40t+36t2.


### 2.3. Results for Third Type of Dominating David Derived Network

In this section, we calculate degree-based topological indices of the dimensiont for third type of Dominating David Derived networks. In the coming theorems, we compute M-Zagreb indices and polynomials.Theorem 9.
Letϒ3≅D3t be the third type of Dominating David Derived network, then the first and second fifth M-Zagreb indices are equal to(54)M1G5ϒ3=8179−469t+360t2,M2G5ϒ3=16720−1763t+1224t2.
(55)M1G5ϒ=∑rs∈EΥSGr−SGs,M1G5ϒ3=6+6E1ϒ3t+6+12E2ϒ3t+6+14E3ϒ3t+8+10E4ϒ3t+8+12E5ϒ3t+8+14E6ϒ3t+10+16E7ϒ3t+12+12E8ϒ3t+12+14E9ϒ3t+12+16E10ϒ3t+14+14E11ϒ3t+14+16E12ϒ3t+16+16E13ϒ3t,=6+64t+6+124t+4+6+144t−4+8+1012t−12+8+1236t2−44t+16+8+144t−4+10+164t−4+12+128t+12+148t−8+12+1636t2−60t+16+14+144t−4+14+164t+4+16+1636t2−76t+40.Proof.
The outcome can be obtained by using the edge partition in Table3. By using equation (1),
By doing some calculations, we get(56)⇒M1G5Υ3=8179−469t+360t2.
Thus, from equation (2),(57)M2G5ϒ=∑rs∈EΥSGr−SGs,M2G5ϒ3=6×6E1ϒ3t+6×12E2ϒ3t+6×14E3ϒ3t+8×10E4ϒ3t+8×12E5ϒ3t+8×14E6ϒ3t+10×16E7ϒ3t+12×12E8ϒ3t+12×14E9ϒ3t+12×16E10ϒ3t+14×14E11ϒ3t+14×16E12ϒ3t+16×16E13ϒ3t,=6×64t+6×124t+4+6×144t−4+8×1012t−12+8×1236t2−44t+16+8×144t−4+10×164t−4+12×128t+12×148t−8+12×1636t2−60t+16+14×144t−4+14×164t+4+16×1636t2−76t+40.
By doing some calculations, we get(58)⇒M2G5ϒ3=16720−1763t+1224t2.Table 3
Edge partition of third type of Dominating David Derived networkD3t based on sum of degrees of end vertices of each edge.
Sr,Ss where rs∈Eϒ3Number of edgesSr,Ss where rs∈Eϒ3Number of edges6,64t12,128t6,124t+412,148t−86,144t−412,1636t2−60t+168,1012t−1214,144t−48,1236t2−44t+1614,164t+48,144t−416,1636t2−76t+4010,164t−4Theorem 10.
Consider the third type of DDD networkϒ3≅D3t for t∈ℕ. Then the first and second general fifth M-Zagreb indices are equal to(59)M1aG5ϒ3=22+a5×21+4a−2×9a+3×10a−11a−3×13a+3×14a+15a+t21+2a3a+6a+4×9a−101+a+11a+3×13a−141+a+15a−19×16a+9t210a+14a+16a,M2aG5ϒ3=23+4a9at+41+a9at+84a4t−4+112a4t−4+160a4t−4+196a4t−4+72a4t+4+224a4t+4+168a8t−8+80a12t−12+256a36t2−76t+40+192a36t2−60t+16+96a36t2−44t+16.Proof.
Letϒ3 be the first type of DDD network. Table 3 shows such an edge partition of D1t. Thus, from equation (3), it follows that(60)M1aG5ϒ=∑rs∈EΥSGr−SGsa.
By using edge partitions in Table3, we get(61)M1aG5ϒ3=6+6aE1ϒ3t+6+12aE2ϒ3t+6+14aE3ϒ3t+8+10aE4ϒ3t+8+12aE5ϒ3t+8+14aE6ϒ3t+10+16aE7ϒ3t+12+12aE8ϒ3t+12+14aE9ϒ3t+12+16aE10ϒ3t+14+14aE11ϒ3t+14+16aE12ϒ3t+16+16aE13ϒ3t,=6+6a4t+6+12a4t+4+6+14a4t−4+8+10a12t−12+8+12a36t2−44t+16+8+14a4t−4+10+16a4t−4+12+12a8t+12+14a8t−8+12+16a36t2−60t+16+14+14a4t−4+14+16a4t+4+16+16a36t2−76t+40.
By doing some calculations, we have(62)⇒M1aG5ϒ3=22+a5×21+4a−2×9a+3×10a−11a−3×13a+3×14a+15a+t21+2a3a+6a+4×9a−101+a+11a+3×13a−141+a+15a−19×16a+9t210a+14a+16a.
From equation (4), we have(63)M2aG5ϒ=∑rs∈EΥSGr−SGsa.
By using edge partitions in Table3, we get(64)M2aG5ϒ3=6×6aE1ϒ3t+6×12aE2ϒ3t+6×14aE3ϒ3t+8×10aE4ϒ3t+8×12aE5ϒ3t+8×14aE6ϒ3t+10×16aE7ϒ3t+12×12aE8ϒ3t+12×14aE9ϒ3t+12×16aE10ϒ3t+14×14aE11ϒ3t+14×16aE12ϒ3t+16×16aE13ϒ3t,=6×6a4t+6×12a4t+4+6×14a4t−4+8×10a12t−12+8×12a36t2−44t+16+8×14a4t−4+10×16a4t−4+12×12a8t+12×14a8t−8+12×16a36t2−60t+16+14×14a4t−4+14×16a4t+4+16×16a36t2−76t+40.
By doing some calculations, we have(65)⇒M2aG5ϒ3=23+4a9at+41+a9at+84a4t−4+112a4t−4+160a4t−4+196a4t−4+72a4t+4+224a4t+4+168a8t−8+80a12t−12+256a36t2−76t+40+192a36t2−60t+16+96a36t2−44t+16.Theorem 11.
Consider the third type of DDD networkϒ3≅D3t for t∈ℕ. Then the first and second hyper fifth M-Zagreb indices are equal to(66)HM12G5ϒ3=48961−2369t+1866t2,HM22G5ϒ3=6445986−102633t+62784t2.Proof.
Letϒ3 be the third type of DDD network. Table 3 shows such an edge partition of D1t. Thus, from equation (5), it follows that(67)HM12G5ϒ=∑rs∈EΥSGr−SGs2.
By using edge partitions in Table3, we get(68)M12G5ϒ3=6+62E1ϒ3t+6+122E2ϒ3t+6+142E3ϒ3t+8+102E4ϒ3t+8+122E5ϒ3t+8+142E6ϒ3t+10+162E7ϒ3t+12+122E8ϒ3t+12+142E9ϒ3t+12+162E10ϒ3t+14+142E11ϒ3t+14+162E12ϒ3t+16+162E13ϒ3t,=6+624t+6+1224t+4+6+1424t−4+8+10212t−12+8+12236t2−44t+16+8+1424t−4+10+1624t−4+12+1228t+12+1428t−8+12+16236t2−60t+16+14+1424t−4+14+1624t+4+16+16236t2−76t+40.
By doing some calculations, we have(69)⇒HM12G5ϒ3=48961−2369t+1866t2.
From equation (6), we have(70)HM22G5ϒ=∑rs∈EΥSGr−SGs2.
By using edge partitions in Table3, we get(71)M22G5ϒ3=6×62E1ϒ3t+6×122E2ϒ3t+6×142E3ϒ3t+8×102E4ϒ3t+8×122E5ϒ3t+8×142E6ϒ3t+10×162E7ϒ3t+12×122E8ϒ3t+12×142E9ϒ3t+12×16aE10ϒ3t+14×142E11ϒ3t+14×162E12ϒ3t+16×162E13ϒ3t,=6×624t+6×1224t+4+6×1424t−4+8×10212t−12+8×12236t2−44t+16+8×1424t−4+10×1624t−4+12×1228t+12×1428t−8+12×16236t2−60t+16+14×1424t−4+14×1624t+4+16×16236t2−76t+40.
By doing some calculations, we have(72)⇒HM22G5ϒ3=6445986−102633t+62784t2.Theorem 12.
Consider the third type of DDD networkϒ3≅D3t for t∈ℕ. Then the third M-Zagreb index is equal to(73)M3G5ϒ3=85−33t+36t2.Proof.
Letϒ3 be the third type of DDD network. Table 3 shows such an edge partition of D3t. Thus from equation (7) it follows that(74)M3G5ϒ=∑rs∈EΥSGr−SGs.
By using edge partitions in Table3, we get(75)M3G5ϒ3=6−6E1ϒ3t+6−12E2ϒ3t+6−14E3ϒ3t+8−10E4ϒ3t+8−12E5ϒ3t+8−14E6ϒ3t+10−16E7ϒ3t+12−12E8ϒ3t+12−14E9ϒ3t+12−16E10ϒ3t+14−14E11ϒ3t+14−16E12ϒ3t+16−16E13ϒ3t+,=6−64t+6−124t+4+6−144t−4+8−1012t−12+8−1236t2−44t+16+8−144t−4+10−164t−4+12−128t+12−148t−8+12−1636t2−60t+16+14−144t−4+14−164t+4+16−1636t2−76t+40.
By doing some calculations, we have(76)⇒M3G5ϒ3=85−33t+36t2.


## 2.1. Results for First Type of Dominating David Derived Networks

In this section, we calculate degree-based topological indices of the dimensiont for first type of Dominating David Derived networks. In the coming theorems, we compute M-Zagreb indices and polynomials.Theorem 1.
Letϒ1≅D1t be the first type of Dominating David Derived network, then the first and second fifth M-Zagreb indices are equal to(8)M1G5ϒ1=1002−2714t+2178t2,M2G5ϒ1=8065−20581t+14697t2.(9)M1G5ϒ=∑rs∈EΥSGr+SGs.M1G5ϒ1=6+6E1Υ1t+6+11E2Υ1t+6+12E3Υ1t+6+14E4Υ1t+7+9E5Υ1t+7+12E6Υ1t+8+11E7Υ1t+8+13E8Υ1t+9+9E9ϒ1t+9+14E10ϒ1t+11+11E11ϒ1t+11+12E12ϒ1t+11+13E13ϒ1t+11+14E14ϒ1t+11+16E15ϒ1t+12+14E16ϒ1t+13+14E17ϒ1t+13+16E18ϒ1t+14+14E19ϒ1t+14+16E20ϒ1t,=6+64t+6+114t+6+124+6+144t−4+7+94t−4+7+124t−4+8+1112t−8+8+134t−4+9+92t−2+9+144t−4+11+119t2−7t+3+11+124+11+134t−4+11+1436t2−68t+32+11+164t−4+12+144t−4+13+144t−4+13+164t−4+14+144t−4+14+1636t2−76t+40.Proof.
The outcome can be obtained by using the edge partition in Table1. By using equation (1),
By doing some calculations, we get(10)⇒M1G5ϒ1=1002−2714t+2178t2.
Thus, from equation (2),(11)M2G5ϒ=∑rs∈EΥSGr+SGs,M2G5ϒ1=6×6E1ϒ1t+6×11E2ϒ1t+6×12E3ϒ1t+6×14E4ϒ1t+7×9E5ϒ1t+7×12E6ϒ1t+8×11E7ϒ1t+8×13E8ϒ1t+9×9E9ϒ1t+9×14E10ϒ1t+11×11E11ϒ1t+11×12E12ϒ1t+11×13E13ϒ1t+11×14E14ϒ1t+11×16E15ϒ1t+12×14E16ϒ1t+13×14E17ϒ1t+13×16E18ϒ1t+14×14E19ϒ1t+14×16E20ϒ1t,=6×64t+6×114t+6×124+6×144t−4+7×94t−4+7×124t−4+8×1112t−8+8×134t−4+9×92t−2+9×144t−4+11×119t2−7t+3+11×124+11×134t−4+11×1436t2−68t+32+11×164t−4+12×144t−4+13×144t−4+13×164t−4+14×144t−4+14×1636t2−76t+40,
By doing some calculations, we get(12)⇒M2G5ϒ1=8065−20581t+14697t2.Table 1
Edge partition of first type of Dominating David Derived networkD1t based on sum of degrees of end vertices of each edge.
Sr,Ss where rs∈Eϒ1Number of edgesSr,Ss where rs∈Eϒ1Number of edges6,64t11,119t2−7t+36,114t11,1246,12411,134t−46,144t−411,1436t2−68t+327,94t−411,164t−47,124t−412,144t−48,1112t−813,144t−48,134t−413,164t−49,92t−214,144t−49,144t−414,1636t2−76t+40Theorem 2.
Consider the first type of DDD networkΥ1≅D1t for t∈ℕ. Then the first and second general fifth M-Zagreb indices are equal to(13)M1aG5ϒ1=−22+3a3a−41+2a−41+a5a+23+a3a51+a−41+a7a−21+a9a+22+a9a−22+a13a−12×19a−4×21a+3×22a+32×25a−8×27a−4×29a+t22+3a3a+3a41+a+41+2a+41+a5a+41+a7a+21+a9a+22+a13a−19×22+a×15a+4×17a+16×19a+4×21a−7×22a+4×23a−68×25a+8×27a+4×29a+t25a62+a+9×22a+36×25a,M2aG5ϒ1=5×23+5a×7a+22+3a9a−41+2a11a−81+a11a−22+3a13a−41+2a13a−23+2a21a−21a22+3a+41+a33a−41+a49a−4×63a−22+a63a+25+a77a−2×81a−22+a91a+121a×3−4×143a+t−19×22+5a×7a+41+a9a+3×22+3a×11a+41+2a11a+22+3a×13a+41+2a13a+23+2a21a+22+3a21a+22+a33a+41+a49a+4×63a+22+a63a−17×22+a×77a+2×81a+22+a91a−7×121a+4×143a+t29×22+5a×7a+9×22+a×77a+9×121a.Proof.
LetΥ1 be the first type of DDD network. Table 1 shows such an edge partition of D1t. Thus, from equation (3), it follows that(14)M1aG5ϒ=∑rs∈EΥSGr+SGsa,
By using edge partitions in Table1, we get(15)M1G5ϒ1=6+6E1Υ1t+6+11E2Υ1t+6+12E3Υ1t+6+14E4Υ1t+7+9E5Υ1t+7+12E6Υ1t+8+11E7Υ1t+8+13E8Υ1t+9+9E9ϒ1t+9+14E10ϒ1t+11+11E11ϒ1t+11+12E12ϒ1t+11+13E13ϒ1t+11+14E14ϒ1t+11+16E15ϒ1t+12+14E16ϒ1t+13+14E17ϒ1t+13+16E18ϒ1t+14+14E19ϒ1t+14+16E20ϒ1t,=6+64t+6+114t+6+124+6+144t−4+7+94t−4+7+124t−4+8+1112t−8+8+134t−4+9+92t−2+9+144t−4+11+119t2−7t+3+11+124+11+134t−4+11+1436t2−68t+32+11+164t−4+12+144t−4+13+144t−4+13+164t−4+14+144t−4+14+1636t2−76t+40,
By doing some calculations, we have(16)⇒M1aG5ϒ1=−22+3a3a−41+2a−41+a5a+23+a3a51+a−41+a7a−21+a9a+22+a9a−22+a13a−12×19a−4×21a+3×22a+32×25a−8×27a−4×29a+t22+3a3a+3a41+a+41+2a+41+a5a+41+a7a+21+a9a+22+a13a−19×22+a×15a+4×17a+16×19a+4×21a−7×22a+4×23a−68×25a+8×27a+4×29a+t25a62+a+9×22a+36×25a,
From equation (4), we have(17)M2aG5ϒ=∑rs∈EΥSGr+SGsa,
By using edge partitions in Table1, we get(18)M2aG5ϒ1=6×62E1ϒ1t+6×112E2ϒ1t+6×122E3ϒ1t+6×142E4ϒ1t+7×92E5ϒ1t+7×122E6ϒ1t+8×112E7ϒ1t+8×132E8ϒ1t+9×92E9ϒ1t+9×142E10ϒ1t+11×112E11ϒ1t+11×122E12ϒ1t+11×132E13ϒ1t+11×142E14ϒ1t+11×162E15ϒ1t+12×142E16ϒ1t+13×142E17ϒ1t+13×162E18ϒ1t+14×142E19ϒ1t+14×162E20ϒ1t,=6×624t+6×1124t+6×1224+6×1424t−4+7×924t−4+7×1224t−4+8×11212t−8+8×1324t−4+9×922t−2+9×1424t−4+11×1129t2−7t+3+11×1224+11×1324t−4+11×14236t2−68t+32+11×1624t−4+12×1424t−4+13×1424t−4+13×1624t−4+14×1424t−4+14×16236t2−76t+40.
By doing some calculations, we have(19)⇒M2aG5ϒ1=5×23+5a×7a+22+3a9a−41+2a11a−81+a11a−22+3a13a−41+2a13a−23+2a21a−21a22+3a+41+a33a−41+a49a−4×63a−22+a63a+25+a77a−2×81a−22+a×91a+121a×3−4×143a+t−19×22+5a×7a+41+a9a+3×22+3a×11a+11a41+2a+22+3a×13a+41+2a13a+23+2a21a+22+3a21a+22+a33a+41+a49a+63a×4+22+a63a−17×22+a×77a+2×81a+22+a91a−7×121a+4×143a+t29×22+5a×7a+9×22+a×77a+9×121a.Theorem 3.
Consider the first type of DDD networkϒ1≅D1t for t∈ℕ. Then, the first and second hyper fifth M-Zagreb indices are equal to(20)HM12G5ϒ1=84005−10286t+7407t2,HM22G5ϒ1=1868329−4442989t+2791881t2.Proof.
Letϒ1 be the first type of DDD network. Table 1 shows such an edge partition of D1t. Thus, from equation (5), it follows that(21)HM12G5ϒ=∑rs∈EΥSGr+SGs2.
By using edge partitions in Table1, we get(22)HM12G5ϒ1=6+62E1Υ1t+6+112E2Υ1t+6+122E3Υ1t+6+142E4Υ1t+7+92E5Υ1t+7+122E6Υ1t+8+112E7Υ1t+8+132E8Υ1t+9+92E9ϒ1t+9+142E10ϒ1t+11+112E11ϒ1t+11+122E12ϒ1t+11+132E13ϒ1t+11+142E14ϒ1t+11+162E15ϒ1t+12+142E16ϒ1t+13+142E17ϒ1t+13+162E18ϒ1t+14+142E19ϒ1t+14+162E20ϒ1t,=6+64t+6+114t+6+124+6+144t−4+7+94t−4+7+124t−4+8+11212t−8+8+1324t−4+9+922t−2+9+1424t−4+11+1129t2−7t+3+11+1224+11+1324t−4+11+14236t2−68t+32+11+1624t−4+12+1424t−4+13+1424t−4+13+1624t−4+14+1424t−4+14+16236t2−76t+40.
By doing some calculations, we have(23)⇒HM12G5ϒ1=84005−10286t+7407t2.
From equation (6), we have(24)HM22G5ϒ=∑rs∈EΥSGr×SGs2.
By using edge partitions in Table1, we get(25)HM22G5ϒ1=6×62E1ϒ1t+6×112E2ϒ1t+6×122E3ϒ1t+6×142E4ϒ1t+7×92E5ϒ1t+7×122E6ϒ1t+8×112E7ϒ1t+8×132E8ϒ1t+9×92E9ϒ1t+9×142E10ϒ1t+11×112E11ϒ1t+11×122E12ϒ1t+11×132E13ϒ1t+11×142E14ϒ1t+11×162E15ϒ1t+12×142E16ϒ1t+13×142E17ϒ1t+13×162E18ϒ1t+14×142E19ϒ1t+14×162E20ϒ1t,=6×624t+6×1124t+6×1224+6×1424t−4+7×924t−4+7×1224t−4+8×11212t−8+8×1324t−4+9×922t−2+9×1424t−4+11×1129t2−7t+3+11×1224+11×1324t−4+11×14236t2−68t+32+11×1624t−4+12×1424t−4+13×1424t−4+13×1624t−4+14×1424t−4+14×16236t2−76t+40.
By doing some calculations, we have(26)⇒HM22G5ϒ1=1868329−4442989t+2791881t2.Theorem 4.
Consider the first type of DDD networkϒ1≅D1t for t∈ℕ. Then, the third M-Zagreb index is equal to(27)M3G5ϒ1=47−37t+45t2.Proof.
Letϒ1 be the first type of DDD network. Table 1 shows such an edge partition of D1t. Thus, from equation (7), it follows that(28)M3G5ϒ=∑rs∈EΥSGr×SGs.
By using edge partitions in Table1, we get(29)M3G5ϒ1=6−6E1ϒ1t+6−11E2ϒ1t+6−12E3ϒ1t+6−14E4ϒ1t+7−9E5ϒ1t+7−12E6ϒ1t+8−11E7ϒ1t+8−13E8ϒ1t+9−9E9ϒ1t+9−14E10ϒ1t+11−11E11ϒ1t+11−12E12ϒ1t+11−13E13ϒ1t+11−14E14ϒ1t+11−16E15ϒ1t+12−14E16ϒ1t+13−14E17ϒ1t+13−16E18ϒ1t+14−14E19ϒ1t+14−16E20ϒ1t+,=6−64t+6−114t+6−124+6−144t−4+7−94t−4+7−124t−4+8−1112t−8+8−134t−4+9−92t−2+9−144t−4+11−119t2−7t+3+11−124+11−134t−4+11−1436t2−68t+32+11−164t−4+12−144t−4+13−144t−4+13−164t−4+14−144t−4+14−1636t2−76t+40.
By doing some calculations, we have(30)⇒M3G5ϒ1=47−37t+45t2.


## 2.2. Results for Second Type of Dominating David Derived Network

Now, we are calculating fifth M-Zagreb topological indices of theϒ2≅D2t, where t∈ℕ for second type of Dominating David Derived network.Theorem 5.
Letϒ2≅D2t be the second type of Dominating David Derived network, then the first and second fifth M-Zagreb indices are equal to(31)M1G5ϒ2=8129−349t+279t2,M2G5ϒ2=24321−823t+591t2.Proof.
The outcome can be obtained by using the edge partition in Table2. By using equation (1),
By doing some calculations, we get(32)⇒M1G5ϒ2=8129−349t+279t2.
Thus, from equation (2),(33)M2G5ϒ=∑rs∈EΥSGr×SGs,M2G5ϒ2=6×6E1ϒ2t+6×8E2ϒ2t+6×10E3ϒ2t+6×11E4ϒ2t+6×12E5ϒ2t+6×14E6ϒ2t+7×8E7ϒ2t+7×12E8ϒ2t+8×11E9ϒ2t+8×13E10ϒ2t+8×14E11ϒ2t+10×11E12ϒ2t+10×13E13ϒ2t+10×14E14ϒ2t+11×12E15ϒ2t+11×14E16ϒ2t+11×16E17ϒ2t+12×14E18ϒ2t+13×14E19ϒ2t+13×16E20ϒ2t+14×14E21ϒ2t+14×16E22ϒ2t,=6×64t+6×84t−4+6×1018t2−30t+14+6×114t+6×124+6×144t−4+7×84t−4+7×124t−4+8×1112t−8+8×134t−4+4t−4+10×118t−4+10×134t−4+10×1436t2−72t+36+11×124+11×144t−4+11×164t−4+12×144t−4+13×144t−4+13×164t−4+14×144t−4+14×1636t2−76t+40.
By doing some calculations, we get(34)⇒M2G5ϒ2=24321−823t+591t2.Table 2
Edge partition of second type of Dominating David Derived networkD2t based on sum of degrees of end vertices of each edge.
Sr,Ss where rs∈Eϒ2Number of edgesSr,Ss where rs∈Eϒ2Number of edges6,64t10,118t−46,84t−410,134t−46,1018t2−30n+1410,1436t2−72t+366,114t11,1246,12411,144t−46,144t−411,164t−47,84t−412,144t−47,124t−413,144t−48,1112t−813,164t−48,134t−414,144t−48,144t−414,1636t2−76t+40(35)M1G5ϒ=∑rs∈EΥSGr×SGs,M1G5ϒ2=6+6E1ϒ2t+6+11E2ϒ2t+6+12E3ϒ2t+6+14E4ϒ2t+7+9E5ϒ2t+7+12E6ϒ2t+8+11E7ϒ2t+8+13E8ϒ2t+9+9E9ϒ2t+9+14E10ϒ2t+11+11E11ϒ2t+11+12E12ϒ2t+11+13E13ϒ2t+11+14E14ϒ2t+11+16E15ϒ2t+12+14E16ϒ2t+13+14E17ϒ2t+13+16E18ϒ2t+14+14E19ϒ2t+14+16E20ϒ2t,=6+64t+6+114t+6+124+6+144t−4+7+94t−4+7+124t−4+8+1112t−8+8+134t−4+9+92t−2+9+144t−4+11+119t2−7t+3+11+124+11+134t−4+11+1436t2−68t+32+11+164t−4+12+144t−4+13+144t−4+13+164t−4+14+144t−4+14+1636t2−76t+40.Theorem 6.
Consider the second type of DDD networkϒ2≅D2t for t∈ℕ. Then the first and second general fifth M-Zagreb indices are equal to(36)M1aG5ϒ2=7×21+4a+22+3a32+a−41+a5a+23+a3a51+a−22+a7a−41+a7a+22+a9a−22+a11a−22+a13a−4×15a−12×19a−8×21a−4×25a−8×27a−4×29a+t−15×21+4a−23+3a32+a+3a41+a+41+a5a+22+a7a+4×31+a7a+41+a7a+22+a11a+22+a13a+4×15a−19×22+a15a+4×17a+16×19a+4×23a+4×25a+8×27a+4×29a+t29×21+4a+22+3a32+a+5a62+a,M2aG5ϒ2=22+3a9a+41+a33a+23+2a21at−1+9×41+a35at−12+t41+a9a+t22+a33a+48a4t−4+56a4t−4+104a4t−4+112a4t−4+130a4t−4+154a4t−4+168a4t−4+176a4t−4+182a4t−4+196a4t−4+208a4t−4+110a8t−4+88a12t−8+60a18t2−30t+14+224a36t2−76t+40.Proof.
Letϒ2 be the second type of DDD network. Table 2 shows such an edge partition of D2t. Thus, from equation (3), it follows that(37)M1aG5ϒ=∑rs∈EΥSGr+SGsa.
By using edge partitions in Table2, we get(38)M1aG5ϒ2=6+6aE1ϒ2t+6+8aE2ϒ2t+6+10aE3ϒ2t+6+11aE4ϒ2t+6+12aE5ϒ2t+6+14aE6ϒ2t+7+8aE7ϒ2t+7+12aE8ϒ2t+8+11aE9ϒ2t+8+13aE10ϒ2t+8+14aE11ϒ2t+10+11aE12ϒ2t+10+13aE13ϒ2t+10+14aE14ϒ2t+11+12aE15ϒ2t+11+14aE16ϒ2t+11+16aE17ϒ2t+12+14aE18ϒ2t+13+14aE19ϒ2t+13+16aE20ϒ2t+14+14aE21ϒ2t+14+16aE22ϒ2t,=6+6a4t+6+8a4t−4+6+10a18t2−30t+14+6+11a4t+6+12a4+6+14a4t−4+7+8a4t−4+7+12a4t−4+8+11a12t−8+8+13a4t−4+8+14a4t−4+10+11a8t−4+10+13a4t−4+10+14a36t2−72t+36+11+12a4+11+14a4t−4+11+16a4t−4+12+14a4t−4+13+14a4t−4+13+16a4t−4+14+14a4t−4+14+16a36t2−76t+40.
By doing some calculations, we have(39)⇒M1aG5ϒ2=7×21+4a+22+3a32+a−41+a5a+23+a3a51+a−22+a7a−41+a7a+22+a9a−22+a11a−22+a13a−4×15a−12×19a−8×21a−4×25a−8×27a−4×29a+t−15×21+4a−23+3a32+a+3a41+a+41+a5a+22+a7a+4×31+a7a+41+a7a+22+a11a+13a22+a+4×15a−19×22+a15a+4×17a+16×19a+4×23a+4×25a+8×27a+4×29a+t29×21+4a+22+3a32+a+5a62+a.
From equation (4), we have(40)M2aG5ϒ=∑rs∈EΥSGr+SGsa.
By using edge partitions in Table2, we get(41)M2G5ϒ2=6×6aE1ϒ2t+6×8aE2ϒ2t+6×10aE3ϒ2t+6×11aE4ϒ2t+6×12aE5ϒ2t+6×14aE6ϒ2t+7×8aE7ϒ2t+7×12aE8ϒ2t+8×11aE9ϒ2t+8×13aE10ϒ2t+8×14aE11ϒ2t+10×11aE12ϒ2t+10×13aE13ϒ2t+10×14aE14ϒ2t+11×12aE15ϒ2t+11×14aE16ϒ2t+11×16aE17ϒ2t+12×14aE18ϒ2t+13×14aE19ϒ2t+13×16aE20ϒ2t+14×14aE21ϒ2t+14×16aE22ϒ2t,=6×6a4t+6×8a4t−4+6×10a18t2−30t+14+6×11a4t+6×12a4+6×14a4t−4+7×8a4t−4+7×12a4t−4+8×11a12t−8+8×13a4t−4+4t−4+10×11a8t−4+10×13a4t−4+10×14a36t2−72t+36+11×12a4+11×14a4t−4+11×16a4t−4+12×14a4t−4+13×14a4t−4+13×16a4t−4+14×14a4t−4+14×16a36t2−76t+40.
By doing some calculations, we have(42)⇒M2aG5ϒ2=22+3a9a+41+a33a+23+2a21at−1+9×41+a35at−12+t41+a9a+t22+a33a+48a4t−4+56a4t−4+104a4t−4+112a4t−4+130a4t−4+154a4t−4+168a4t−4+176a4t−4+182a4t−4+196a4t−4+208a+4t−4110a8t−4+88a12t−8+60a18t2−30t+14+224a36t2−76t+40.Theorem 7.
Consider the second type of DDD networkϒ2≅D2t for t∈ℕ. Then the first and second hyper fifth M-Zagreb indices are equal to(43)HM12G5ϒ2=83875−9985t+7218t2,HM22G5ϒ2=3253531−127816t+80523t2.Proof.
Letϒ2 be the second type of DDD network. Table 2 shows such an edge partition of D2t. Thus, from equation (5), it follows that(44)HM12G5ϒ=∑rs∈EΥSGr+SGs2.
By using edge partitions in Table2, we get(45)M12G5ϒ2=6+62E1ϒ2t+6+82E2ϒ2t+6+102E3ϒ2t+6+112E4ϒ2t+6+122E5ϒ2t+6+142E6ϒ2t+7+82E7ϒ2t+7+122E8ϒ2t+8+112E9ϒ2t+8+132E10ϒ2t+8+142E11ϒ2t+10+112E12ϒ2t+10+132E13ϒ2t+10+142E14ϒ2t+11+122E15ϒ2t+11+142E16ϒ2t+11+162E17ϒ2t+12+142E18ϒ2t+13+142E19ϒ2t+13+162E20ϒ2t+14+142E21ϒ2t+14+162E22ϒ2t,=6+624t+6+824t−4+6+10218t2−30t+14+6+1124t+6+1224+6+1424t−4+7+824t−4+7+1224t−4+8+11212t−8+8+1324t−4+8+1424t−4+10+1128t−4+10+1324t−4+10+14236t2−72t+36+11+1224+11+1424t−4+11+1624t−4+12+1424t−4+13+1424t−4+13+1624t−4+14+1424t−4+14+16236t2−76t+40.
By doing some calculations, we have(46)⇒HM12G5ϒ2=83875−9985t+7218t2.
From equation (6), we have(47)HM22G5ϒ=∑rs∈EΥSGr+SGs2.
By using edge partitions in Table2, we get(48)M22G5ϒ2=6×62E1ϒ2t+6×82E2ϒ2t+6×102E3ϒ2t+6×112E4ϒ2t+6×122E5ϒ2t+6×142E6ϒ2t+7×82E7ϒ2t+7×122E8ϒ2t+8×112E9ϒ2t+8×132E10ϒ2t+8×142E11ϒ2t+10×112E12ϒ2t+10×132E13ϒ2t+10×142E14ϒ2t+11×122E15ϒ2t+11×142E16ϒ2t+11×162E17ϒ2t+12×142E18ϒ2t+13×14aE19ϒ2t+13×162E20ϒ2t+14×142E21ϒ2t+14×162E22ϒ2t,=6×624t+6×824t−4+6×10a18t2−30t+14+6×1124t+6×1224+6×1424t−4+7×824t−4+7×1224t−4+8×11212t−8+8×1324t−4+4t−4+10×1128t−4+10×1324t−4+10×14236t2−72t+36+11×1224+11×1424t−4+11×1624t−4+12×1424t−4+13×1424t−4+13×1624t−4+14×1424t−4+14×16236t2−76t+40.
By doing some calculations, we have(49)⇒HM22G5ϒ2=3253531−127816t+80523t2.Theorem 8.
Consider the second type of DDD networkΥ2≅D2t for t∈ℕ. Then the third M-Zagreb index is equal to(50)M3G5ϒ2=813−40t+36t2.Proof.
LetΥ2 be the second type of DDD network. Table 2 shows such an edge partition of D2t. Thus, from equation (7), it follows that(51)M3G5ϒ=∑rs∈EΥSGr+SGs.
By using edge partitions in Table2, we get(52)M3G5Υ2=|6−6E1Υ2t+6−8E2Υ2t+6−10E3Υ2t+6−11E4Υ2t+6−12E5Υ2t+6−14E6Υ2t+7−8E7Υ2t+7−12E8Υ2t+8−11E9Υ2t+8−13E10Υ2t+8−14E11Υ2t+10−11E12Υ2t+10−13E13Υ2t+10−14E14Υ2t+11−12E15Υ2t+11−14E16Υ2t+11−16E17Υ2t+12−14E18Υ2t+13−14E19Υ2t+13−16E20Υ2t+14−14E21Υ2t+14−16E22Υ2t+,=6−64t+|6−84t−4+6−1018t2−30t+14+6−114t+6−124+6−144t−4+7−84t−4+7−124t−4+8−1112t−8+8−134t−4+8−144t−4+|10−11|8t−4+10−134t−4+10−1436t2−72t+36+411−12+11−144t−4+11−164t−4+12−144t−4+13−144t−4+13−164t−4+14−144t−4+|14−16|36t2−76t+40.
By doing some calculations, we have(53)⇒M3G5ϒ2=813−40t+36t2.


## 2.3. Results for Third Type of Dominating David Derived Network

In this section, we calculate degree-based topological indices of the dimensiont for third type of Dominating David Derived networks. In the coming theorems, we compute M-Zagreb indices and polynomials.Theorem 9.
Letϒ3≅D3t be the third type of Dominating David Derived network, then the first and second fifth M-Zagreb indices are equal to(54)M1G5ϒ3=8179−469t+360t2,M2G5ϒ3=16720−1763t+1224t2.
(55)M1G5ϒ=∑rs∈EΥSGr−SGs,M1G5ϒ3=6+6E1ϒ3t+6+12E2ϒ3t+6+14E3ϒ3t+8+10E4ϒ3t+8+12E5ϒ3t+8+14E6ϒ3t+10+16E7ϒ3t+12+12E8ϒ3t+12+14E9ϒ3t+12+16E10ϒ3t+14+14E11ϒ3t+14+16E12ϒ3t+16+16E13ϒ3t,=6+64t+6+124t+4+6+144t−4+8+1012t−12+8+1236t2−44t+16+8+144t−4+10+164t−4+12+128t+12+148t−8+12+1636t2−60t+16+14+144t−4+14+164t+4+16+1636t2−76t+40.Proof.
The outcome can be obtained by using the edge partition in Table3. By using equation (1),
By doing some calculations, we get(56)⇒M1G5Υ3=8179−469t+360t2.
Thus, from equation (2),(57)M2G5ϒ=∑rs∈EΥSGr−SGs,M2G5ϒ3=6×6E1ϒ3t+6×12E2ϒ3t+6×14E3ϒ3t+8×10E4ϒ3t+8×12E5ϒ3t+8×14E6ϒ3t+10×16E7ϒ3t+12×12E8ϒ3t+12×14E9ϒ3t+12×16E10ϒ3t+14×14E11ϒ3t+14×16E12ϒ3t+16×16E13ϒ3t,=6×64t+6×124t+4+6×144t−4+8×1012t−12+8×1236t2−44t+16+8×144t−4+10×164t−4+12×128t+12×148t−8+12×1636t2−60t+16+14×144t−4+14×164t+4+16×1636t2−76t+40.
By doing some calculations, we get(58)⇒M2G5ϒ3=16720−1763t+1224t2.Table 3
Edge partition of third type of Dominating David Derived networkD3t based on sum of degrees of end vertices of each edge.
Sr,Ss where rs∈Eϒ3Number of edgesSr,Ss where rs∈Eϒ3Number of edges6,64t12,128t6,124t+412,148t−86,144t−412,1636t2−60t+168,1012t−1214,144t−48,1236t2−44t+1614,164t+48,144t−416,1636t2−76t+4010,164t−4Theorem 10.
Consider the third type of DDD networkϒ3≅D3t for t∈ℕ. Then the first and second general fifth M-Zagreb indices are equal to(59)M1aG5ϒ3=22+a5×21+4a−2×9a+3×10a−11a−3×13a+3×14a+15a+t21+2a3a+6a+4×9a−101+a+11a+3×13a−141+a+15a−19×16a+9t210a+14a+16a,M2aG5ϒ3=23+4a9at+41+a9at+84a4t−4+112a4t−4+160a4t−4+196a4t−4+72a4t+4+224a4t+4+168a8t−8+80a12t−12+256a36t2−76t+40+192a36t2−60t+16+96a36t2−44t+16.Proof.
Letϒ3 be the first type of DDD network. Table 3 shows such an edge partition of D1t. Thus, from equation (3), it follows that(60)M1aG5ϒ=∑rs∈EΥSGr−SGsa.
By using edge partitions in Table3, we get(61)M1aG5ϒ3=6+6aE1ϒ3t+6+12aE2ϒ3t+6+14aE3ϒ3t+8+10aE4ϒ3t+8+12aE5ϒ3t+8+14aE6ϒ3t+10+16aE7ϒ3t+12+12aE8ϒ3t+12+14aE9ϒ3t+12+16aE10ϒ3t+14+14aE11ϒ3t+14+16aE12ϒ3t+16+16aE13ϒ3t,=6+6a4t+6+12a4t+4+6+14a4t−4+8+10a12t−12+8+12a36t2−44t+16+8+14a4t−4+10+16a4t−4+12+12a8t+12+14a8t−8+12+16a36t2−60t+16+14+14a4t−4+14+16a4t+4+16+16a36t2−76t+40.
By doing some calculations, we have(62)⇒M1aG5ϒ3=22+a5×21+4a−2×9a+3×10a−11a−3×13a+3×14a+15a+t21+2a3a+6a+4×9a−101+a+11a+3×13a−141+a+15a−19×16a+9t210a+14a+16a.
From equation (4), we have(63)M2aG5ϒ=∑rs∈EΥSGr−SGsa.
By using edge partitions in Table3, we get(64)M2aG5ϒ3=6×6aE1ϒ3t+6×12aE2ϒ3t+6×14aE3ϒ3t+8×10aE4ϒ3t+8×12aE5ϒ3t+8×14aE6ϒ3t+10×16aE7ϒ3t+12×12aE8ϒ3t+12×14aE9ϒ3t+12×16aE10ϒ3t+14×14aE11ϒ3t+14×16aE12ϒ3t+16×16aE13ϒ3t,=6×6a4t+6×12a4t+4+6×14a4t−4+8×10a12t−12+8×12a36t2−44t+16+8×14a4t−4+10×16a4t−4+12×12a8t+12×14a8t−8+12×16a36t2−60t+16+14×14a4t−4+14×16a4t+4+16×16a36t2−76t+40.
By doing some calculations, we have(65)⇒M2aG5ϒ3=23+4a9at+41+a9at+84a4t−4+112a4t−4+160a4t−4+196a4t−4+72a4t+4+224a4t+4+168a8t−8+80a12t−12+256a36t2−76t+40+192a36t2−60t+16+96a36t2−44t+16.Theorem 11.
Consider the third type of DDD networkϒ3≅D3t for t∈ℕ. Then the first and second hyper fifth M-Zagreb indices are equal to(66)HM12G5ϒ3=48961−2369t+1866t2,HM22G5ϒ3=6445986−102633t+62784t2.Proof.
Letϒ3 be the third type of DDD network. Table 3 shows such an edge partition of D1t. Thus, from equation (5), it follows that(67)HM12G5ϒ=∑rs∈EΥSGr−SGs2.
By using edge partitions in Table3, we get(68)M12G5ϒ3=6+62E1ϒ3t+6+122E2ϒ3t+6+142E3ϒ3t+8+102E4ϒ3t+8+122E5ϒ3t+8+142E6ϒ3t+10+162E7ϒ3t+12+122E8ϒ3t+12+142E9ϒ3t+12+162E10ϒ3t+14+142E11ϒ3t+14+162E12ϒ3t+16+162E13ϒ3t,=6+624t+6+1224t+4+6+1424t−4+8+10212t−12+8+12236t2−44t+16+8+1424t−4+10+1624t−4+12+1228t+12+1428t−8+12+16236t2−60t+16+14+1424t−4+14+1624t+4+16+16236t2−76t+40.
By doing some calculations, we have(69)⇒HM12G5ϒ3=48961−2369t+1866t2.
From equation (6), we have(70)HM22G5ϒ=∑rs∈EΥSGr−SGs2.
By using edge partitions in Table3, we get(71)M22G5ϒ3=6×62E1ϒ3t+6×122E2ϒ3t+6×142E3ϒ3t+8×102E4ϒ3t+8×122E5ϒ3t+8×142E6ϒ3t+10×162E7ϒ3t+12×122E8ϒ3t+12×142E9ϒ3t+12×16aE10ϒ3t+14×142E11ϒ3t+14×162E12ϒ3t+16×162E13ϒ3t,=6×624t+6×1224t+4+6×1424t−4+8×10212t−12+8×12236t2−44t+16+8×1424t−4+10×1624t−4+12×1228t+12×1428t−8+12×16236t2−60t+16+14×1424t−4+14×1624t+4+16×16236t2−76t+40.
By doing some calculations, we have(72)⇒HM22G5ϒ3=6445986−102633t+62784t2.Theorem 12.
Consider the third type of DDD networkϒ3≅D3t for t∈ℕ. Then the third M-Zagreb index is equal to(73)M3G5ϒ3=85−33t+36t2.Proof.
Letϒ3 be the third type of DDD network. Table 3 shows such an edge partition of D3t. Thus from equation (7) it follows that(74)M3G5ϒ=∑rs∈EΥSGr−SGs.
By using edge partitions in Table3, we get(75)M3G5ϒ3=6−6E1ϒ3t+6−12E2ϒ3t+6−14E3ϒ3t+8−10E4ϒ3t+8−12E5ϒ3t+8−14E6ϒ3t+10−16E7ϒ3t+12−12E8ϒ3t+12−14E9ϒ3t+12−16E10ϒ3t+14−14E11ϒ3t+14−16E12ϒ3t+16−16E13ϒ3t+,=6−64t+6−124t+4+6−144t−4+8−1012t−12+8−1236t2−44t+16+8−144t−4+10−164t−4+12−128t+12−148t−8+12−1636t2−60t+16+14−144t−4+14−164t+4+16−1636t2−76t+40.
By doing some calculations, we have(76)⇒M3G5ϒ3=85−33t+36t2.


## 3. Comparison of Indices

This section contains the comparison of indices and graphical details of different types of DDD networks as given in Figures6–14.Figure 6
Comparison betweenM1G5ϒ1andM2G5ϒ1.Figure 7
Comparison betweenHM12G5ϒ1 and HM22G5ϒ1.Figure 8
Graphical detail ofM3G5ϒ1.Figure 9
Comparison betweenM1G5ϒ2 and M2G5ϒ2.Figure 10
Comparison betweenHM12G5ϒ2andHM22G5ϒ2.Figure 11
Graphical detail ofM3G5ϒ2.Figure 12
Comparison betweenM1G5ϒ3andM2G5ϒ3.Figure 13
Comparison betweenHM12G5ϒ3andHM22G5ϒ3.Figure 14
Graphical detail ofM3G5ϒ3.


## 4. Conclusion

In this manuscript, we computed sum of degree-based indices for some derived graphs of honeycomb structure. We also computed certain sum of degree-based polynomials such as fifth M-Zagreb, fifth hyper M-Zagreb, generalized fifth M-Zagreb indices for all types of Dominating David Derived networks and we also provide comparison of indies in form of graphs, all the graphs are increasing, increase with the value oft. These facts may be useful for people working in computer science and chemistry who encounter honeycomb networks. These results can also play a vital part in the determination of the significance of honeycomb derived networks. As other topological indices, determining the representations of derived graphs like these, is an open question for many other topological indices.Honeycomb structure is present in different products of sports, aerospace, woodworking and loudspeaker technology due to its physical and chemical properties. Today, mathematicians are working on derived structures of honeycomb to enhance its physical and chemical properties. In future, for more stable structures we will be able to derive some new graphs of honeycomb structure and will find their physical and chemical properties via topological indices.



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