# Eccentricity-Based Topological Invariants of First Type of Dominating David-Derived Networks

Authors: Haidar Ali; Ayesha Umer; Parvez Ali; Mohamed Sesay
Journal: Journal of Mathematics
 (2023)

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## Abstract

Previous research has shown a substantial correlation between a chemical compound’s molecular structure and its chemical characteristics. Understanding the physical characteristics and biological functions of biomolecules can be improved with the aid of topological indices based on their molecular structure. Eccentricity-connectivity indices, which describe molecular structures based on distance, have been applied to the mathematical modelling of a wide range of biological activities. In this article, we compute the exact formulae of various versions of the eccentricity-connectivity index for first type of Dominating David-derived network. The findings can be applied to computer-aided molecular design methods in pharmaceutical engineering.


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## Body

## 1. Introduction and Preliminary Results

Cheminformatics is a new field that combines chemistry, mathematics, and information science. The use of graph theory in chemistry is widespread. Chemical graph theory is a topology branch of mathematical chemistry that uses graph theory to model chemical occurrences mathematically. There has been a lot of research done in this area over the last few decades. This theory plays an important role in the field of chemical sciences. Some references are provided, which should demonstrate the significance of this field [1, 2].To identify the graph, a numeric number, a polynomial, an integer, a sequence, or a matrix might all be utilised. A topological graph is defined by a numeric value known as a distance-based topological index, which remains constant during the graph automorphism process and characterises the topology of the graph [3, 4].We have a honeycomb networkHC that is n-dimensional. Add a new vertex to divide the HCn−edges in half. An edge should connect the new vertices in each hexagon cell if they are four units apart within the cell. Vertices should be added at new edge intersections. The essential vertices and edges of the honeycomb network must be removed. Divide each horizontal edge in half by adding a new vertex. The evolving graph is known as the Dominating David-derived network. The Dominating David-derived network of the first type designed by D1n, of dimension n, may be constructed by linking vertices of degree two of DDDn as an edge that is not in the border [5].LetG^=V^,E^ be a graph with V^ is a collection of nonempty vertices, and E^ is a set of edges. There are numerous topological indices for a connected graph that may be used to investigate chemical compounds. This hypothesis has had a profound influence on the advancement of chemical research by few references [6, 7]. If p^, q^ belongs to V^G^, the distance dp^,q^ between p^ and q^ is equal to the length of any shortest path in G^ linking p^ and q^. Eccentricity is defined mathematically as the greatest distance between p^∈G^ and any vertex q^∈G^ [8]. In a mathematical style,
(1)ξp^=maxdp^,q^∣p^∈V^G^.The total eccentricity index [9] is presented as
(2)£G^=∑s∈V^G^ξs,where ξs represent the eccentricity of the vertex (s) by Farooq and Malik [10].The average eccentricityavecG^ of a graph is
(3)avecG^=1ℓ∑s∈V^G^ξs,where ℓ denote the total number of vertices by following references [2, 11–13].The Zagreb indexMG^ of a graph is
(4)MG^=∑s∈V^G^ξs2.


## 2. Main Results

The first type of Dominating David-derived networkD1n for all, even dimension is discussed in this section, as well as close equations for several topological indices. Here you can find the empirically terminate consequences for total eccentricity index, average eccentricity index, and eccentricity-based Zagreb index [14]. The order and size of D1n is VD1n=45n2−39n+12 and ED1n=81n2−170n+369+170k, where range of k is 0.5n−2. To study more about eccentricity indices, see [15–18].


### 2.1. Result on Dominating David-Derived Network of Type 1

Dominating David-derived networks constructed from a honeycomb structure are essential for chemical research into materials with low density and high compressibility. The stress in other aerospace-related materials is also studied using these structures.D12 is displayed in Figure 1. The vertex partition of D1n is displayed in Tables 1 and 2. We process eccentricity ξp^ and total eccentricity Zagreb index.Figure 1
Dominating David-derived network of the first typeD12.Table 1
Vertex partition ofD1n.
Sets
ξ
s
Vertices
Range;n≡0 (mod 4)
V
^
1
5
n
+
k
+
2
p
4
n
k
=
n
−
2; 0≤p≤3n−2
V
^
2
4
n
+
k
+
2
p
6
n
k
=
2
n
−
3; 1≤p≤3n−2
V
^
3
6
n
−
1
+
4
p
2
1
≤
p
≤
1.5
n
−
2
V
^
4
6
n
+
2
p
n
−
k
+
4
p
0
≤
p
≤
0.75
n
−
2; k=n−6
V
^
5
6
n
−
3
+
2
p
6
p
1
≤
p
≤
0.75
n
−
1
V
^
6
7.5
n
−
2
+
2
p
3
n
−
6
−
12
p
0
≤
p
≤
0.25
n
−
1
V
^
7
7.5
n
−
3
+
2
p
4.5
n
−
8
−
18
p
0
≤
p
≤
0.25
n
−
1
V
^
8
6
n
−
3
3
n
−
2
V
^
9
6
n
−
2
2Table 2
Vertex partition ofD1n.
Sets
ξ
s
Vertices
Range;n≡2 (mod 4)
V
^
1
5
n
+
k
+
2
p
4
n
k
=
n
−
2; 0≤p≤3n−2
V
^
2
4
n
+
k
+
2
p
6
n
k
=
2
n
−
3; 1≤p≤3n−2
V
^
3
6
n
−
1
+
4
p
2
0
≤
p
≤
1.5
n
−
2
V
^
4
6
n
−
3
+
2
p
6
p
1
≤
p
≤
0.75
n
−
0.5
V
^
5
6
n
+
2
p
n
−
k
+
4
p
0
≤
p
≤
0.75
n
−
2.5; k=n−6
V
^
6
7.5
n
−
3
+
2
p
3
n
−
12
p
0
≤
p
≤
0.25
n
−
0.5
V
^
7
7.5
n
−
2
+
2
p
4.5
n
+
1
−
18
p
+
1
0
≤
p
≤
0.25
n
−
1.5
V
^
8
6
n
−
3
3
n
−
2
V
^
9
6
n
−
2
2


### 2.2. Case I: Whenn≡0 (Mod 4)

The eccentricity of four vertices is6n+2p+k for 0≤p≤0.75n−2, where 0≤k≤6n−12−8p. The eccentricity of four vertices is 6n+1+2p+4k for 0≤p≤0.75n−2, where 0≤k≤1.5n−4−r.Theorem 1.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡0 (mod 4), the total eccentricity index is
(5)£G^=296.875n3−245.75n2+6n+18.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡0 (mod 4) to evaluate the total eccentricity index as follows, utilized the vertices partitioned from Table 1 and Equation (2).
(6)£G^=∑s∈V^G^ξs,£G^=∑s∈V^1G^ξs+∑s∈V^2G^ξs+∑s∈V^3G^ξs+∑s∈V^4G^ξs+∑s∈V^5G^ξs+∑s∈V^6G^ξs+∑s∈V^7G^ξs+ξV^8+ξV^9,⇒£G^=∑P=03n−2.∑k=n−2n−25n+k+2p4n+∑p=13n−2.∑k=2n−32n−34n+k+2p6n+∑p=11.5n−26n−1+4p2+∑p=00.75n−2.∑k=n−6n−66n+2pn−k+4p+∑p=10.75n−16n−3+2p6p+∑p=00.25n−17.5n−2+2p3n−6−12p+∑p=00.25n−17.5n−3+2p4.5n−8−18p+6n−33n−2+6n−22.
After calculations, we have(7)⇒£G^=296.875n3−245.75n2+6n+18.Theorem 2.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡0 (mod 4), the average eccentricity index is
(8)avecD1n=145n2−39n+12296.875n3−245.75n2+6n+18.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡0 (mod 4) to evaluate the average eccentricity index as follows, utilized the vertices partitioned from Table 1 and Equation (3).
(9)avecG^=1ℓ∑s∈V^G^ξs,⇒avecG^=1ℓ∑s∈V^1G^ξs+1ℓ∑s∈V^2G^ξs+1ℓ∑s∈V^3Gξs+1ℓ∑s∈V^4G^ξs+1ℓ∑s∈V^5G^ξs+1ℓ∑s∈V^6G^ξs+1ℓ∑s∈V^7G^ξs+1ℓξV^8+1ℓξV^9,⇒avecG^=145n2−39n+12∑P=03n−2.∑k=n−2n−25n+k+2p4n+∑p=13n−2.∑k=2n−32n−34n+k+2p6n+∑p=11.5n−26n−1+4p2+∑p=00.75n−2.∑k=n−6n−66n+2pn−k+4p+∑p=10.75n−16n−3+2p6p+∑p=00.25n−17.5n−2+2p3n−6−12p+6n−33n−2+∑p=00.25n−17.5n−3+2p4.5n−8−18p+6n−22.
After calculations, we have(10)⇒avecG^=145n2−39n+12296.875n3−245.75n2+6n+18.Theorem 3.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡0 (mod 4), the Zagreb eccentricity index is
(11)MG^=2713.281250n4−3521.416667n3+1160.75n2+86.6666667n−70.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡0 (mod 4) to evaluate the Zagreb eccentricity index as follows, utilized the vertices partitioned from Table 1 and Equation (4).
(12)MG^=∑s∈V^G^ξs2,⇒MG^=∑s∈V^1G^ξs2+∑s∈V^2G^ξs2+∑s∈V^3G^ξs2+∑s∈V^4G^ξs2+∑s∈V^5G^ξs2+∑s∈V^6G^ξs2+∑s∈V^7G^ξs2+ξV∧82+ξV∧92,⇒MG^=∑P=03n−2.∑k=n−2n−25n+k+2p24n+∑p=13n−2.∑k=2n−32n−34n+k+2p26n+∑p=11.5n−26n−1+4p22+∑p=00.75n−2.∑k=n−6n−66n+2p2n−k+4p+∑p=10.75n−16n−3+2p26p+∑p=00.25n−17.5n−2+2p23n−6−12p+∑p=00.25n−17.5n−3+2p24.5n−8−18p+6n−323n−2+6n−222.
After calculations, we have(13)⇒MG^=2713.281250n4−3521.416667n3+1160.75n2+86.6666667n−70.


### 2.3. Case II: Whenn≡2 (Mod 4)

The eccentricity of four vertices is6n+2p+k for 0≤p≤0.75n−1.5, where 0≤k≤6n−12−8p. The eccentricity of four vertices is 6n+1+2p+4k for 0≤p≤0.75n−2.5, where 0≤k≤1.5n−4−2p.Theorem 4.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡2 (mod 4), the total eccentricity index is
(14)£G^=296.875n3−245.75n2+40.5n+5.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡2 (mod 4) to evaluate the total eccentricity index as follows, utilized the vertices partitioned from Table 2 and Equation (2).
(15)£G^=∑s∈V^G^ξs,⇒£G^=∑s∈V^1G^ξs+∑s∈V^2G^ξs+∑s∈V^3G^ξs+∑s∈V^4G^ξs+∑s∈V^5G^ξs+∑s∈V^6G^ξs+∑s∈V^7G^ξs+ξV^8+ξV^9,⇒£G^=∑P=03n−2.∑k=n−2n−25n+k+2p4n+∑p=13n−2.∑k=2n−32n−34n+k+2p6n+∑p=01.5n−26n−1+4p2+∑p=10.75n−0.56n−3+2p6p+∑p=00.75n−2.5.∑k=n−6n−66n+2pn−k+4p+∑p=00.25n−0.57.5n−3+2p3n−12p+∑p=00.25n−1.57.5n−2+2p4.5n+1−18p+1+6n−33n−2+6n−22.
After calculations, we have(16)£G^=296.8750n3−245.75n2+40.5n+5.Theorem 5.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡2 (mod 4), the average eccentricity index is
(17)avecD1n=145n2−39n+12296.875n3−245.75n2+40.5n+5.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡2 (mod 4) to evaluate the average eccentricity index as follows, utilized the vertex partitions from Table 2 and Equation (3).
(18)avecG^=1ℓ∑s∈V^G^ξs,⇒avecG^=1ℓ∑s∈V^1G^ξs+1ℓ∑s∈V^2G^ξs+1ℓ∑s∈V^3G^ξs+1ℓ∑s∈V^4G^ξs+1ℓ∑s∈V^5G^ξs+1ℓ∑s∈V^6G^ξs+1ℓ∑s∈V^7G^ξs+1ℓξV^8+1ℓξV^9,⇒avecG^=145n2−39n+12∑P=03n−2.∑k=n−2n−25n+k+2p4n+∑p=13n−2.∑k=2n−32n−34n+k+2p6n+∑p=01.5n−26n−1+4p2+∑p=00.25n−0.57.5n−3+2p3n−12p+∑p=10.75n−0.56n−3+2p6p+∑p=00.75n−2.5.∑k=n−6n−66n+2pn−k+4p+∑p=00.25n−1.57.5n−2+2p4.5n+1−18p+1+6n−33n−2+6n−22.
After calculations, we have(19)⇒avecg^=145n2−39n+12296.875n3−245.75n2+40.5n+5.Theorem 6.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡2 (mod 4), the Zagreb eccentricity index is
(20)MG^=2713.281250n4−3521.416667n3+1401.5n2−102.3333333n−28.5.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡2 (mod 4) to evaluate the Zagreb eccentricity index as follows, utilized the vertices partitioned from Table 2 and Equation (4).
(21)MG^=∑s∈V^G^ξs2,⇒MG^=∑s∈V^1G^ξs2+∑s∈V^2G^ξs2+∑s∈V^3G^ξs2+∑s∈V^4G^ξs2+∑s∈V^5G^ξs2+∑s∈V^6G^ξs2+∑s∈V^7G^ξs2+ξV∧82+ξV∧92,⇒MG^=∑P=03n−2.∑k=n−2n−25n+k+2p24n+∑p=13n−2.∑k=2n−32n−34n+k+2p26n+∑p=01.5n−26n−1+4p22+∑p=10.75n−0.56n−3+2p26p+∑p=00.75n−2.5.∑k=n−6n−66n+2p2n−k+4p+∑p=00.25n−0.57.5n−3+2p23n−12p+∑p=00.25n−1.57.5n−2+2p24.5n+1−18p+1+6n−323n−2+6n−222.
After calculations, we have(22)⇒MG^=2713.281250n4−3521.416667n3+1401.500000n2−102.3333333n−28.5.


## 2.1. Result on Dominating David-Derived Network of Type 1

Dominating David-derived networks constructed from a honeycomb structure are essential for chemical research into materials with low density and high compressibility. The stress in other aerospace-related materials is also studied using these structures.D12 is displayed in Figure 1. The vertex partition of D1n is displayed in Tables 1 and 2. We process eccentricity ξp^ and total eccentricity Zagreb index.Figure 1
Dominating David-derived network of the first typeD12.Table 1
Vertex partition ofD1n.
Sets
ξ
s
Vertices
Range;n≡0 (mod 4)
V
^
1
5
n
+
k
+
2
p
4
n
k
=
n
−
2; 0≤p≤3n−2
V
^
2
4
n
+
k
+
2
p
6
n
k
=
2
n
−
3; 1≤p≤3n−2
V
^
3
6
n
−
1
+
4
p
2
1
≤
p
≤
1.5
n
−
2
V
^
4
6
n
+
2
p
n
−
k
+
4
p
0
≤
p
≤
0.75
n
−
2; k=n−6
V
^
5
6
n
−
3
+
2
p
6
p
1
≤
p
≤
0.75
n
−
1
V
^
6
7.5
n
−
2
+
2
p
3
n
−
6
−
12
p
0
≤
p
≤
0.25
n
−
1
V
^
7
7.5
n
−
3
+
2
p
4.5
n
−
8
−
18
p
0
≤
p
≤
0.25
n
−
1
V
^
8
6
n
−
3
3
n
−
2
V
^
9
6
n
−
2
2Table 2
Vertex partition ofD1n.
Sets
ξ
s
Vertices
Range;n≡2 (mod 4)
V
^
1
5
n
+
k
+
2
p
4
n
k
=
n
−
2; 0≤p≤3n−2
V
^
2
4
n
+
k
+
2
p
6
n
k
=
2
n
−
3; 1≤p≤3n−2
V
^
3
6
n
−
1
+
4
p
2
0
≤
p
≤
1.5
n
−
2
V
^
4
6
n
−
3
+
2
p
6
p
1
≤
p
≤
0.75
n
−
0.5
V
^
5
6
n
+
2
p
n
−
k
+
4
p
0
≤
p
≤
0.75
n
−
2.5; k=n−6
V
^
6
7.5
n
−
3
+
2
p
3
n
−
12
p
0
≤
p
≤
0.25
n
−
0.5
V
^
7
7.5
n
−
2
+
2
p
4.5
n
+
1
−
18
p
+
1
0
≤
p
≤
0.25
n
−
1.5
V
^
8
6
n
−
3
3
n
−
2
V
^
9
6
n
−
2
2


## 2.2. Case I: Whenn≡0 (Mod 4)

The eccentricity of four vertices is6n+2p+k for 0≤p≤0.75n−2, where 0≤k≤6n−12−8p. The eccentricity of four vertices is 6n+1+2p+4k for 0≤p≤0.75n−2, where 0≤k≤1.5n−4−r.Theorem 1.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡0 (mod 4), the total eccentricity index is
(5)£G^=296.875n3−245.75n2+6n+18.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡0 (mod 4) to evaluate the total eccentricity index as follows, utilized the vertices partitioned from Table 1 and Equation (2).
(6)£G^=∑s∈V^G^ξs,£G^=∑s∈V^1G^ξs+∑s∈V^2G^ξs+∑s∈V^3G^ξs+∑s∈V^4G^ξs+∑s∈V^5G^ξs+∑s∈V^6G^ξs+∑s∈V^7G^ξs+ξV^8+ξV^9,⇒£G^=∑P=03n−2.∑k=n−2n−25n+k+2p4n+∑p=13n−2.∑k=2n−32n−34n+k+2p6n+∑p=11.5n−26n−1+4p2+∑p=00.75n−2.∑k=n−6n−66n+2pn−k+4p+∑p=10.75n−16n−3+2p6p+∑p=00.25n−17.5n−2+2p3n−6−12p+∑p=00.25n−17.5n−3+2p4.5n−8−18p+6n−33n−2+6n−22.
After calculations, we have(7)⇒£G^=296.875n3−245.75n2+6n+18.Theorem 2.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡0 (mod 4), the average eccentricity index is
(8)avecD1n=145n2−39n+12296.875n3−245.75n2+6n+18.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡0 (mod 4) to evaluate the average eccentricity index as follows, utilized the vertices partitioned from Table 1 and Equation (3).
(9)avecG^=1ℓ∑s∈V^G^ξs,⇒avecG^=1ℓ∑s∈V^1G^ξs+1ℓ∑s∈V^2G^ξs+1ℓ∑s∈V^3Gξs+1ℓ∑s∈V^4G^ξs+1ℓ∑s∈V^5G^ξs+1ℓ∑s∈V^6G^ξs+1ℓ∑s∈V^7G^ξs+1ℓξV^8+1ℓξV^9,⇒avecG^=145n2−39n+12∑P=03n−2.∑k=n−2n−25n+k+2p4n+∑p=13n−2.∑k=2n−32n−34n+k+2p6n+∑p=11.5n−26n−1+4p2+∑p=00.75n−2.∑k=n−6n−66n+2pn−k+4p+∑p=10.75n−16n−3+2p6p+∑p=00.25n−17.5n−2+2p3n−6−12p+6n−33n−2+∑p=00.25n−17.5n−3+2p4.5n−8−18p+6n−22.
After calculations, we have(10)⇒avecG^=145n2−39n+12296.875n3−245.75n2+6n+18.Theorem 3.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡0 (mod 4), the Zagreb eccentricity index is
(11)MG^=2713.281250n4−3521.416667n3+1160.75n2+86.6666667n−70.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡0 (mod 4) to evaluate the Zagreb eccentricity index as follows, utilized the vertices partitioned from Table 1 and Equation (4).
(12)MG^=∑s∈V^G^ξs2,⇒MG^=∑s∈V^1G^ξs2+∑s∈V^2G^ξs2+∑s∈V^3G^ξs2+∑s∈V^4G^ξs2+∑s∈V^5G^ξs2+∑s∈V^6G^ξs2+∑s∈V^7G^ξs2+ξV∧82+ξV∧92,⇒MG^=∑P=03n−2.∑k=n−2n−25n+k+2p24n+∑p=13n−2.∑k=2n−32n−34n+k+2p26n+∑p=11.5n−26n−1+4p22+∑p=00.75n−2.∑k=n−6n−66n+2p2n−k+4p+∑p=10.75n−16n−3+2p26p+∑p=00.25n−17.5n−2+2p23n−6−12p+∑p=00.25n−17.5n−3+2p24.5n−8−18p+6n−323n−2+6n−222.
After calculations, we have(13)⇒MG^=2713.281250n4−3521.416667n3+1160.75n2+86.6666667n−70.


## 2.3. Case II: Whenn≡2 (Mod 4)

The eccentricity of four vertices is6n+2p+k for 0≤p≤0.75n−1.5, where 0≤k≤6n−12−8p. The eccentricity of four vertices is 6n+1+2p+4k for 0≤p≤0.75n−2.5, where 0≤k≤1.5n−4−2p.Theorem 4.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡2 (mod 4), the total eccentricity index is
(14)£G^=296.875n3−245.75n2+40.5n+5.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡2 (mod 4) to evaluate the total eccentricity index as follows, utilized the vertices partitioned from Table 2 and Equation (2).
(15)£G^=∑s∈V^G^ξs,⇒£G^=∑s∈V^1G^ξs+∑s∈V^2G^ξs+∑s∈V^3G^ξs+∑s∈V^4G^ξs+∑s∈V^5G^ξs+∑s∈V^6G^ξs+∑s∈V^7G^ξs+ξV^8+ξV^9,⇒£G^=∑P=03n−2.∑k=n−2n−25n+k+2p4n+∑p=13n−2.∑k=2n−32n−34n+k+2p6n+∑p=01.5n−26n−1+4p2+∑p=10.75n−0.56n−3+2p6p+∑p=00.75n−2.5.∑k=n−6n−66n+2pn−k+4p+∑p=00.25n−0.57.5n−3+2p3n−12p+∑p=00.25n−1.57.5n−2+2p4.5n+1−18p+1+6n−33n−2+6n−22.
After calculations, we have(16)£G^=296.8750n3−245.75n2+40.5n+5.Theorem 5.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡2 (mod 4), the average eccentricity index is
(17)avecD1n=145n2−39n+12296.875n3−245.75n2+40.5n+5.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡2 (mod 4) to evaluate the average eccentricity index as follows, utilized the vertex partitions from Table 2 and Equation (3).
(18)avecG^=1ℓ∑s∈V^G^ξs,⇒avecG^=1ℓ∑s∈V^1G^ξs+1ℓ∑s∈V^2G^ξs+1ℓ∑s∈V^3G^ξs+1ℓ∑s∈V^4G^ξs+1ℓ∑s∈V^5G^ξs+1ℓ∑s∈V^6G^ξs+1ℓ∑s∈V^7G^ξs+1ℓξV^8+1ℓξV^9,⇒avecG^=145n2−39n+12∑P=03n−2.∑k=n−2n−25n+k+2p4n+∑p=13n−2.∑k=2n−32n−34n+k+2p6n+∑p=01.5n−26n−1+4p2+∑p=00.25n−0.57.5n−3+2p3n−12p+∑p=10.75n−0.56n−3+2p6p+∑p=00.75n−2.5.∑k=n−6n−66n+2pn−k+4p+∑p=00.25n−1.57.5n−2+2p4.5n+1−18p+1+6n−33n−2+6n−22.
After calculations, we have(19)⇒avecg^=145n2−39n+12296.875n3−245.75n2+40.5n+5.Theorem 6.
Consider a graphD1n of dimension n, ∀n∈ℕ at n≡2 (mod 4), the Zagreb eccentricity index is
(20)MG^=2713.281250n4−3521.416667n3+1401.5n2−102.3333333n−28.5.Proof.
ConsiderD1n of dimension n, ∀n∈ℕ, n≡2 (mod 4) to evaluate the Zagreb eccentricity index as follows, utilized the vertices partitioned from Table 2 and Equation (4).
(21)MG^=∑s∈V^G^ξs2,⇒MG^=∑s∈V^1G^ξs2+∑s∈V^2G^ξs2+∑s∈V^3G^ξs2+∑s∈V^4G^ξs2+∑s∈V^5G^ξs2+∑s∈V^6G^ξs2+∑s∈V^7G^ξs2+ξV∧82+ξV∧92,⇒MG^=∑P=03n−2.∑k=n−2n−25n+k+2p24n+∑p=13n−2.∑k=2n−32n−34n+k+2p26n+∑p=01.5n−26n−1+4p22+∑p=10.75n−0.56n−3+2p26p+∑p=00.75n−2.5.∑k=n−6n−66n+2p2n−k+4p+∑p=00.25n−0.57.5n−3+2p23n−12p+∑p=00.25n−1.57.5n−2+2p24.5n+1−18p+1+6n−323n−2+6n−222.
After calculations, we have(22)⇒MG^=2713.281250n4−3521.416667n3+1401.500000n2−102.3333333n−28.5.


## 3. Conclusion

Dominating David-derived networks illustrated from honeycomb formation have strong influence on research in this paper. In the study of chemical compounds, materials have little density and excessive compression. These formations are then used to study strain in various aircraft material structures. As a result, we conduct significant subject studies that will be useful for biochemical organic web research. To investigate the first type of Dominating David-derived networkD1n, topological indices of eccentricity based on network, total eccentricity £G^, average eccentricity avecG^, and Zagreb index MG^ were used. The creation of new networks and architectures is something we are interested in doing in the future. After that, we will investigate their topological indices, which will be very useful in understanding the underlying topologies.



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*Source: 7562648-2023-02-23.xml*