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Symmetry 2018, 10(11), 619; https://doi.org/10.3390/sym10110619

Article
On Degree-Based Topological Indices of Symmetric Chemical Structures
1
School of Mathematics and Physics, Anhui Jianzhu University, Hefei 230601, China
2
Department of Mathematics, Government College University, Faisalabad 38000, Pakistan
*
Author to whom correspondence should be addressed.
Received: 18 October 2018 / Accepted: 6 November 2018 / Published: 9 November 2018

Abstract

:
A Topological index also known as connectivity index is a type of a molecular descriptor that is calculated based on the molecular graph of a chemical compound. Topological indices are numerical parameters of a graph which characterize its topology and are usually graph invariant. In QSAR/QSPR study, physico-chemical properties and topological indices such as Randić, atom-bond connectivity (ABC) and geometric-arithmetic (GA) index are used to predict the bioactivity of chemical compounds. Graph theory has found a considerable use in this area of research. In this paper, we study HDCN1(m,n) and HDCN2(m,n) of dimension m , n and derive analytical closed results of general Randić index R α ( G ) for different values of α . We also compute the general first Zagreb, ABC, GA, A B C 4 and G A 5 indices for these Hex derived cage networks for the first time and give closed formulas of these degree-based indices.
Keywords:
general randić index; atom-bond connectivity (ABC) index; geometric-arithmetic (GA) index; Hex-Derived Cage networks; HDCN1(m,n), HDCN2(m,n)

1. Introduction

A graph is formed by vertices and edges connecting the vertices. A network is a connected simple graph having no multiple edges and loops. A topological index is a function Top : ∑ → R where R is the set of real numbers and ∑ is the finite simple graph with property that Top ( G 1 ) = T o p ( G 2 ) if G 1 and G 2 are isomorphic. A topological index is a numerical value associated with chemical constitution for correlation of chemical structure with various physical properties, chemical reactivity or biological activity. Many tools, such as topological indices has provided by graph theory to the chemists. Cheminformatics is new subject which is a combination of chemistry, mathematics and information science. It studies Quantitative structure-activity (QSAR) and structure-property (QSPR) relationships that are used to predict the biological activities and properties of chemical compounds. In the QSAR /QSPR study, physico-chemical properties and topological indices such as Wiener index, Szeged index, Randić index, Zagreb indices and A B C index are used to predict bioactivity of the chemical compounds. “In terms of graph theory, the structural formula of a chemical compound represents the molecular graph, in which vertices are represents to atoms and edges as chemical bonds”. A molecular descriptor is a numeric number, which represents the properties of a chemical graph. Basically, a molecular descriptor and topological descriptor are different from each other. A molecular descriptor represents the underlying chemical graph but a topological descriptor are the representation of physico-chemical properties of underlying chemical graph in addition to show the whole structure. Topological indices have many applications in the field of nanobiotechnology and QSAR/QSPR study. Topological indices were firstly introduced by Wiener [1], he named the resulting index as path number while he was working on boiling point of Paraffin. Later on, it renamed as Wiener index [2]. Consider “n” Hex-Derived networks ( H D N 1 ( 1 , 1 ) ) , ( H D N 1 ( 2 , 2 ) ) and so on to ( H D N 1 ( m , n ) ) . Connect every boundary vertices of ( H D N 1 ( 1 , 1 ) ) to its mirror image vertices in ( H D N 1 ( 2 , 2 ) ) by an edge and so on to ( H D N 1 ( m , n ) ) . As a result, we found a graph, which is called Hex-Derived Cage networks with “n” layers. In this article, the notations which we used take from the books [3,4].
In this article, Graph ( G ) is considered to be a graph with vertex set V ( G ) and edge set E ( G ) , the d ( a ) is the degree of vertex a V ( G ) and S ( a ) = b N G ( a ) d ( b ) where N G ( a ) = { b V ( G ) a b E ( G ) } .
Let G be a graph. Then the Wiener index is written as
W ( G ) = 1 2 ( a , b ) d ( a , b )
The Randić index [5] is the oldest degree-based topological index invented by Milan Randić, denoted as R 1 2 ( G ) and defined as
R 1 2 ( G ) = a b E ( G ) 1 d ( a ) d ( b )
R α ( G ) is a general Randić index and it is defined as
R α ( G ) = a b E ( G ) ( d ( a ) d ( b ) ) α for α R
A topological index which has a great importance was introduced by Ivan Gutman and T r i n a j s t i c ´ is Zagreb index and defined as
M 1 ( G ) = a b E ( G ) ( d ( a ) + d ( b ) )
Estrada et al. in [6] invented a very famous degree-based topological index ABC and defined as
A B C ( G ) = a b E ( G ) d ( a ) + d ( b ) 2 d ( a ) d ( b )
GA index is also a very famous connectivity topological descriptor, which invented by Vukičević et al. [7] and denoted as
G A ( G ) = a b E ( G ) 2 d ( a ) d ( b ) ( d ( a ) + d ( b ) )
A B C 4 and G A 5 indices find only if we find the edge partition of interconnection networks each edge in the graphs depend on sum of the degrees of end vertices. A B C 4 index invented by Ghorbani et al. [8] and written as
A B C 4 ( G ) = a b E ( G ) S ( a ) + S ( b ) 2 S ( a ) S ( b )
The latest version of index is G A 5 invented by Graovac et al. [9] and defined as
G A 5 ( G ) = a b E ( G ) 2 S ( a ) S ( b ) ( S ( a ) + S ( b ) )
For any graph G for α = 1 , the general Randić index is second Zagreb index.

2. Main Results

Hex-Derived Cage networks H D C N 1 ( m , n ) (show in Figure 1) and H D C N 2 ( m , n ) (show in Figure 2) give closed formulas of that indices, we study the general Randić, first Zagreb, ABC, GA, A B C 4 and G A 5 indices of certain graphs in [10]. These days there is a broad research activity on A B C and G A indices and their variants, for additionally investigation of topological indices of different families see, [1,11,12,13,14,15,16,17,18,19,20,21,22,23].

2.1. Results for Hex-Derived Cage Networks

We compute specific degree-based topological indices of Hex-Derived Cage networks. In this paper, we calculate Randić index R α ( G ) with α = 1 , 1 , 1 2 , 1 2 , M 1 , ABC, GA, A B C 4 and G A 5 for Hex-Derived Cage networks H D C N 1 ( m , n ) and H D C N 2 ( m , n ) .
Theorem 1.
Let G 1 H D C N 1 ( m , n ) be the Hex-Derived Cage network, then its general Randić index is equal to
R α ( G 1 ) = 18 ( 108 n 3 219 n 2 + 25 n + 91 ) , α = 1 ; 6 ( 36 n 3 + 3 ( 7 3 34 ) n 2 + ( 4 21 + 6 7 + 28 6 84 3 + 12 2 + 35 ) n + 2 42 8 21 12 7 56 6 + 100 3 + 39 ) , α = 1 2 ; 11907 n 3 17003 n 2 + 12343 n 3051 21168 , α = 1 ; 15 n 3 4 + ( 8 3 125 12 ) n 2 + ( 4 7 + 4 6 32 3 + 2 + 5 3 7 + 109 21 ) n 8 7 8 6 + 38 3 + 3 2 + 2 6 7 10 3 7 + 13 14 , α = 1 2 .
Proof. 
Let G 1 be the Hex-Derived Cage network ( H D C N 1 ( m , n ) ) where m = n 5 . The edge set of H D C N 1 ( m , n ) are divided into seventeen partitions based on the degree of end vertices shows in Table 1. Thus from Equation ( 3 ) , is follows that
R α ( G 1 ) = a b E ( G ) ( d ( a ) d ( b ) ) α
For α = 1
R 1 ( G 1 ) = j = 1 17 a b E j ( G ) d e g ( u ) · d e g ( v )
By using the edge partition given in Table 1, we have R 1 ( G 1 ) = 18 | E 1 ( G 1 ) | + 21 | E 2 ( G 1 ) | + 24 | E 3 ( G 1 ) | + 27 | E 4 ( G 1 ) | + 36 | E 5 ( G 1 ) | + 42 | E 6 ( G 1 ) | + 48 | E 7 ( G 1 ) | + 72 | E 8 ( G 1 ) | + 49 | E 9 ( G 1 ) | + 63 | E 10 ( G 1 ) | + 84 | E 11 ( G 1 ) | + 64 | E 12 ( G 1 ) | + 72 | E 13 ( G 1 ) | + 96 | E 14 ( G 1 ) | + 81 | E 15 ( G 1 ) | + 108 | E 16 ( G 1 ) | + 144 | E 17 ( G 1 ) |
After simplification, we have
R 1 ( G 1 ) = 18 ( 108 n 3 219 n 2 + 25 n + 91 )
For α = 1 2
R 1 2 ( G 1 ) = j = 1 17 a b E j ( G ) d ( a ) · d ( b )
Using the edge partition from Table 1, we have R 1 2 ( G 1 ) = 3 2 | E 1 ( G 1 ) | + 21 | E 2 ( G 1 ) | + 2 6 | E 3 ( G 1 ) | + 3 3 | E 4 ( G 1 ) | + 6 | E 5 ( G 1 ) | + 42 | E 6 ( G 1 ) | + 4 3 | E 7 ( G 1 ) | + 6 2 | E 8 ( G 1 ) | + 7 | E 9 ( G 1 ) | + 3 7 | E 10 ( G 1 ) | + 2 21 | E 11 ( G 1 ) | + 8 | E 12 ( G 1 ) | + 6 2 | E 13 ( G 1 ) | + 4 6 | E 14 ( G 1 ) | + 9 | E 15 ( G 1 ) | + 6 3 | E 16 ( G 1 ) | + 12 | E 17 ( G 1 ) |
After simplification, we have R 1 2 ( G 1 ) = 6 ( 36 n 3 + 3 ( 7 3 34 ) n 2 + ( 4 21 + 6 7 + 28 6 84 3 + 12 2 + 35 ) n + 2 42 8 21 12 7 56 6 + 100 3 + 39 ) For α = 1
R 1 ( G 1 ) = j = 1 17 a b E j ( G ) 1 d ( a ) · d ( b )
R 1 ( G 1 ) = 1 18 | E 1 ( G 1 ) | + 1 21 | E 2 ( G 1 ) | + 1 24 | E 3 ( G 1 ) | + 1 27 | E 4 ( G 1 ) | + 1 36 | E 5 ( G 1 ) | + 1 42 | E 6 ( G 1 ) | + 1 48 | E 7 ( G 1 ) | + 1 72 | E 8 ( G 1 ) | + 1 49 | E 9 ( G 1 ) | + 1 63 | E 10 ( G 1 ) | + 1 84 | E 11 ( G 1 ) | + 1 64 | E 12 ( G 1 ) | + 1 72 | E 13 ( G 1 ) | + 1 96 | E 14 ( G 1 ) | + 1 81 | E 15 ( G 1 ) | + 1 108 | E 16 ( G 1 ) | + 1 144 | E 17 ( G 1 ) |
After simplification, we have
R 1 ( G 1 ) = 11907 n 3 17003 n 2 + 12343 n 3051 21168
For α = 1 2
R 1 2 ( G 1 ) = j = 1 17 a b E j ( G ) 1 d ( a ) · d ( b )
R 1 2 ( G 1 ) = 2 6 | E 1 ( G 1 ) | + 21 21 | E 2 ( G 1 ) | + 6 12 | E 3 ( G 1 ) | + 3 9 | E 4 ( G 1 ) | + 1 6 | E 5 ( G 1 ) | + 42 42 | E 6 ( G 1 ) | + 3 12 | E 7 ( G 1 ) | + 2 12 | E 8 ( G 1 ) | + 1 7 | E 9 ( G 1 ) | + 7 21 | E 10 ( G 1 ) | + 21 42 | E 11 ( G 1 ) | + 1 8 | E 12 ( G 1 ) | + 2 12 | E 13 ( G 1 ) | + 6 24 | E 14 ( G 1 ) | + 1 9 | E 15 ( G 1 ) | + 3 18 | E 16 ( G 1 ) | + 1 12 | E 17 ( G 1 ) |
After simplification, we have R 1 2 ( G 1 ) = 15 n 3 4 + ( 8 3 125 12 ) n 2 + ( 4 7 + 4 6 32 3 + 2 + 5 3 7 + 109 21 ) n 8 7 8 6 + 38 3 + 3 2 + 2 6 7 10 3 7 + 13 14  □
In the below theorem, we calculate the Zagreb index of G 1 (m,n).
Theorem 2.
The first Zagreb index of hex-derived cage network H D C N 1 ( m , n ) is equal to
M 1 ( G 1 ) = 18 ( 27 n 3 51 n 2 + 10 n + 14 )
Proof. 
With the help of Table 1, we calculate the Zagreb index as
M 1 ( G 1 ) = a b E ( G ) ( d ( a ) + d ( b ) ) = j = 1 17 a b E j ( G ) ( d ( a ) + d ( b ) )
M 1 ( G 1 ) = 9 | E 1 ( G 1 ) | + 10 | E 2 ( G 1 ) | + 11 | E 3 ( G 1 ) | + 12 | E 4 ( G 1 ) | + 15 | E 5 ( G 1 ) | + 13 | E 6 ( G 1 ) | + 14 | E 7 ( G 1 ) | + 18 | E 8 ( G 1 ) | + 14 | E 9 ( G 1 ) | + 16 | E 10 ( G 1 ) | + 19 | E 11 ( G 1 ) | + 16 | E 12 ( G 1 ) | + 17 | E 13 ( G 1 ) | + 20 | E 14 ( G 1 ) | + 18 | E 15 ( G 1 ) | + 21 | E 16 ( G 1 ) | + 24 | E 17 ( G 1 ) |
After some calculations, we get
M 1 ( G 1 ) = 18 ( 27 n 3 51 n 2 + 10 n + 14 )
 □
In the next theorem, we calculate the A B C , G A , A B C 4 and G A 5 indices of Hex-Derived Cage network H D C N 1 ( m , n ) .
Theorem 3.
Let HDCN1(m,n) be Hex-Derived Cage network, then we have
  • A B C ( G 1 ) = 3 4 ( 4 13 + 22 ) n 3 + 1 12 ( 8 57 + 24 30 33 22 108 13 + 64 ) n 2 + ( 80 3 + 8 6 7 + 4 2 + 54 3 7 + 3 7 2 + 5 11 2 + 9 6 8 19 3 + 51 7 + 7 13 7 30 ) n + 44 16 6 7 4 2 120 3 7 7 2 18 6 + 8 19 3 2 51 7 + 2 66 7 + 6 30 .
  • G A ( G 1 ) ) = 117 n 3 5 + 3 35 ( 185 3 749 ) n 2 + ( 108 5 + 144 2 17 444 3 7 + 1248 6 55 + 9 7 2 + 348 21 95 ) n + 24 42 13 696 21 95 9 7 2496 6 55 + 540 3 7 + 120 2 17 + 18 .
  • A B C 4 ( G 1 ) = 6 5 38 7 ( n 4 ) 2 + 2 69 79 ( n 2 ) + 1 3 62 5 n ( 3 n 2 15 n + 19 ) + 1 15 89 2 n ( 3 n 2 17 n + 24 ) + 177 14 ( n 2 5 n + 6 ) + 2 115 77 ( n 2 5 n + 6 ) + 2 86 105 ( n 2 5 n + 6 ) + 2 2 5 ( n 2 5 n + 6 ) + 1 7 83 2 ( n 2 6 n + 8 ) + 8 23 34 ( n 2 9 n + 20 ) + 2 113 79 ( n 2 ) + 334 395 ( n 2 ) + 4 30 79 ( n 2 ) + 6 15 79 ( n 2 ) + 6 58 41 ( n 3 ) + 4 34 41 ( n 3 ) + 12 7 41 ( n 3 ) + 18 6 287 ( n 3 ) + 20 ( n 4 ) 2 253 + 2 194 115 ( n 4 ) 2 + 2 151 161 ( n 4 ) 2 + 144 ( n 4 ) 5293 + 48 ( n 4 ) 85 + 3 5 254 79 ( n 4 ) + 54 41 2 ( n 4 ) + 2 114 67 ( n 4 ) + 2 447 469 ( n 4 ) + 3 22 29 ( n 4 ) + 4 6 23 ( n 4 ) + 12 6 29 ( n 4 ) + 6 93 469 ( n 4 ) + 12 138 1189 ( n 4 ) + 30 2 119 ( n 4 ) + 18 5 391 ( n 4 ) + 28 6 737 ( n 4 ) + 3 17 134 ( n 5 ) + 6 29 114 ( n 5 ) + 24 67 33 ( n 5 ) + 21 25 2 ( n 5 ) + 206 194 385 + 4 678 11 7 + 6 41 7 + 24 29 + 2 110 19 + 3 43 14 + 3 53 19 + 4 138 77 + 4 94 55 + 4 786 737 + 4 78 79 + 4 74 77 + 2 82 95 + 6 190 287 + 24 22 41 7 + 6 65 133 + 60 3 7 7 + 12 10 41 + 6 115 779 + 12 10 77 + 12 34 287 + 12 2 17 + 12 78 779 + 24 5 91 + 60 2 247 + 36 2 553 .
  • G A 5 ( G 1 ) = 40 13 14 ( n 4 ) 2 + 48 163 1659 ( n 2 ) + 12 7 10 n ( 3 n 2 15 n + 19 ) + 24 29 210 ( n 2 5 n + 6 ) + 16 13 77 ( n 2 5 n + 6 ) + 12 19 70 ( n 2 5 n + 6 ) + 18 5 21 ( n 2 5 n + 6 ) + 9 n 3 33 n 2 66 n + 3 14 2607 ( n 2 ) + 36 169 790 ( n 2 ) + 48 107 553 ( n 2 ) + 144 115 79 ( n 2 ) + 24 55 574 ( n 3 ) + 72 43 205 ( n 3 ) + 216 59 82 ( n 3 ) + 64 19 41 ( n 3 ) + 6 17 253 ( n 4 ) 2 + 8 11 230 ( n 4 ) 2 + 16 17 161 ( n 4 ) 2 + 12 73 5293 ( n 4 ) + 3 17 4623 ( n 4 ) + 6 25 2211 ( n 4 ) + 24 97 2010 ( n 4 ) + 48 151 1407 ( n 4 ) + 24 35 1189 ( n 4 ) + 48 137 1173 ( n 4 ) + 8 21 986 ( n 4 ) + 48 101 561 ( n 4 ) + 48 49 510 ( n 4 ) + 48 95 469 ( n 4 ) + 48 43 406 ( n 4 ) + 24 19 357 ( n 4 ) + 20 43 158 ( n 4 ) + 40 39 134 ( n 4 ) + 32 15 29 ( n 4 ) + 12 ( n 4 ) + 36 ( n 5 ) + 48 5214 145 + 48 4422 133 + 48 1558 79 + 48 1122 67 + 48 1066 67 + 16 779 39 + 96 574 97 + 16 494 17 + 48 462 47 + 32 287 23 + 16 266 11 + 96 231 61 + 32 203 19 + 72 190 83 + 48 154 25 + 96 91 41 + 21 79 16 + 336 66 115 + 3 55 + 28 41 15 + 32 38 9 + 36 19 7 + 192 7 11 + 1336 2 33 + 258 .
Proof. 
From Table 1 we calculate the A B C ( G 1 ) as
A B C ( G 1 ) = a b E ( G ) d ( a ) + d ( b ) 2 d ( a ) · d ( b ) = j = 1 17 a b E j ( G ) d ( a ) + d ( b ) 2 d ( a ) · d ( b )
A B C ( G 1 ) = 14 6 | E 1 ( G 1 ) | + 2 42 21 | E 2 ( G 1 ) | + 6 4 | E 3 ( G 1 ) | + 30 9 | E 4 ( G 1 ) | + 13 6 | E 5 ( G 1 ) | + 462 42 | E 6 ( G 1 ) | + 1 2 | E 7 ( G 1 ) | + 2 3 | E 8 ( G 1 ) | + 2 3 7 | E 9 ( G 1 ) | + 2 3 | E 10 ( G 1 ) | + 357 42 | E 11 ( G 1 ) | + 14 8 | E 12 ( G 1 ) | + 30 12 | E 13 ( G 1 ) | + 3 4 | E 14 ( G 1 ) | + 4 9 | E 15 ( G 1 ) | + 57 18 | E 16 ( G 1 ) | + 22 12 | E 17 ( G 1 ) | .
After simplification, we have A B C ( G 1 ) = 3 4 ( 4 13 + 22 ) n 3 + 1 12 ( 8 57 + 24 30 33 22 108 13 + 64 ) n 2 + ( 80 3 + 8 6 7 + 4 2 + 54 3 7 + 3 7 2 + 5 11 2 + 9 6 8 19 3 + 51 7 + 7 13 7 30 ) n + 44 16 6 7 4 2 120 3 7 7 2 18 6 + 8 19 3 2 51 7 + 2 66 7 + 6 30 .
Now we calculate G A from Equation (6) as
G A ( G 1 ) = a b E ( G ) 2 d ( a ) d ( b ) ( d ( a ) + d ( b ) ) = j = 1 17 a b E j ( G ) 2 d ( a ) d ( b ) ( d ( a ) + d ( b ) )
From Table 1 calculate G A ( G 1 ) as
G A ( G 1 ) = 2 2 3 | E 1 ( G 1 ) | + 21 5 | E 2 ( G 1 ) | + 4 6 11 | E 3 ( G 1 ) | + 3 2 | E 4 ( G 1 ) | + 4 5 | E 5 ( G 1 ) | + 2 42 13 | E 6 ( G 1 ) | + 4 3 7 | E 7 ( G 1 ) | + 2 2 3 | E 8 ( G 1 ) | + 1 | E 9 ( G 1 ) | + 3 7 8 | E 10 ( G 1 ) | + 4 21 19 | E 11 ( G 1 ) | + 1 | E 12 ( G 1 ) | + 12 2 17 | E 13 ( G 1 ) | + 2 6 5 | E 14 ( G 1 ) | + 1 | E 15 ( G 1 ) | + 4 3 7 | E 16 ( G 1 ) | + 1 | E 17 ( G 1 ) | .
After simplification, we have
G A ( G 1 ) = 117 n 3 5 + 3 35 ( 185 3 749 ) n 2 + ( 108 5 + 144 2 17 444 3 7 + 1248 6 55 + 9 7 2 + 348 21 95 ) n + 24 42 13 696 21 95 9 7 2496 6 55 + 540 3 7 + 120 2 17 + 18 .
If we consider an edge partition based on degree sum of neighbors of end vertices; then the edge set E ( H D C N 1 ( m , n ) ) are divided into sixtynine edge partition E j ( H D C N 1 ( m , n ) ) , 18 j 86 shows in Table 2.
From Equation (7), we have
A B C 4 ( G 1 ) = a b E ( G ) S ( a ) + S ( b ) 2 S ( a ) S ( b ) = j = 18 86 a b E j ( G ) S ( a ) + S ( b ) 2 S ( a ) S ( b ) .
From Table 2 we use edge partition, we get
A B C 4 ( G 1 ) = 2 1066 67 | E 18 ( G 1 ) | + 1456 41 | E 19 ( G 1 ) | + 1976 51 | E 20 ( G 1 ) | + 2 1372 77 | E 21 ( G 1 ) | + 1400 39 | E 22 ( G 1 ) | + 1568 42 | E 23 ( G 1 ) | + 1624 43 | E 24 ( G 1 ) | + 1848 47 | E 25 ( G 1 ) | + 2 1876 95 | E 26 ( G 1 ) | + 2 2212 107 | E 27 ( G 1 ) | + 2296 55 | E 28 ( G 1 ) | + 1980 48 | E 29 ( G 1 ) | + 2 2010 97 | E 30 ( G 1 ) | + 2040 49 | E 31 ( G 1 ) | + 2 2070 99 | E 32 ( G 1 ) | + 2520 57 | E 33 ( G 1 ) | + 1792 44 | E 34 ( G 1 ) | + 1856 45 | E 35 ( G 1 ) | + 2432 54 | E 36 ( G 1 ) | + 2624 57 | E 37 ( G 1 ) | + 2 2178 99 | E 38 ( G 1 ) | + 2211 50 | E 39 ( G 1 ) | + 2 2244 101 | E 40 ( G 1 ) | + 2277 51 | E 41 ( G 1 ) | + 2607 3 | E 56 ( G 1 ) | + 2 2772 117 | E 43 ( G 1 ) | + 2736 56 | E 44 ( G 1 ) | + 2 2844 115 | E 45 ( G 1 ) | + 2952 59 | E 46 ( G 1 ) | + 3024 60 | E 47 ( G 1 ) | + 3240 63 | E 48 ( G 1 ) | + 2009 45 | E 49 ( G 1 ) | + 2 2296 97 | E 50 ( G 1 ) | + 2 3116 117 | E 51 ( G 1 ) | + 2 2450 99 | E 52 ( G 1 ) | + 2 3234 115 | E 53 ( G 1 ) | + 3871 64 | E 54 ( G 1 ) | + 1 | E 55 ( G 1 ) | + 2 3350 117 | E 56 ( G 1 ) | + 2 3950 129 | E 57 ( G 1 ) | + 3248 57 | E 58 ( G 1 ) | + 3696 56 | E 59 ( G 1 ) | + 4256 66 | E 60 ( G 1 ) | + 4292 69 | E 61 ( G 1 ) | + 3364 58 | E 62 ( G 1 ) | + 3944 63 | E 63 ( G 1 ) | + 4756 70 | E 64 ( G 1 ) | + 2 4422 133 | E 65 ( G 1 ) | + 4488 67 | E 66 ( G 1 ) | + 2 5214 145 | E 67 ( G 1 ) | + 5544 75 | E 68 ( G 1 ) | + 4489 67 | E 69 ( G 1 ) | + 4623 68 | E 70 ( G 1 ) | + 5293 73 | E 71 ( G 1 ) | + 2 5628 151 | E 72 ( G 1 ) | + 4624 68 | E 73 ( G 1 ) | + 2 4 692 137 | E 74 ( G 1 ) | + 5712 76 | E 75 ( G 1 ) | + 4761 69 | E 76 ( G 1 ) | + 2 5796 153 | E 77 ( G 1 ) | + 6232 79 | E 78 ( G 1 ) | + 6840 83 | E 79 | ( G 1 ) + 2 6636 163 | E 80 ( G 1 ) | + 2 7110 169 | E 81 ( G 1 ) | + 4724 82 | E 82 ( G 1 ) | + 7380 86 | E 83 ( G 1 ) | + 7056 84 | E 84 ( G 1 ) | + 7560 87 | E 85 ( G 1 ) | + 1 | E 86 ( G 1 ) | .
After simplification, we get
A B C 4 ( G 1 ) = 6 5 38 7 ( n 4 ) 2 + 2 69 79 ( n 2 ) + 1 3 62 5 n ( 3 n 2 15 n + 19 ) + 1 15 89 2 n ( 3 n 2 17 n + 24 ) + 177 14 ( n 2 5 n + 6 ) + 2 115 77 ( n 2 5 n + 6 ) + 2 86 105 ( n 2 5 n + 6 ) + 2 2 5 ( n 2 5 n + 6 ) + 1 7 83 2 ( n 2 6 n + 8 ) + 8 23 34 ( n 2 9 n + 20 ) + 2 113 79 ( n 2 ) + 334 395 ( n 2 ) + 4 30 79 ( n 2 ) + 6 15 79 ( n 2 ) + 6 58 41 ( n 3 ) + 4 34 41 ( n 3 ) + 12 7 41 ( n 3 ) + 18 6 287 ( n 3 ) + 20 ( n 4 ) 2 253 + 2 194 115 ( n 4 ) 2 + 2 151 161 ( n 4 ) 2 + 144 ( n 4 ) 5293 + 48 ( n 4 ) 85 + 3 5 254 79 ( n 4 ) + 54 41 2 ( n 4 ) + 2 114 67 ( n 4 ) + 2 447 469 ( n 4 ) + 3 22 29 ( n 4 ) + 4 6 23 ( n 4 ) + 12 6 29 ( n 4 ) + 6 93 469 ( n 4 ) + 12 138 1189 ( n 4 ) + 30 2 119 ( n 4 ) + 18 5 391 ( n 4 ) + 28 6 737 ( n 4 ) + 3 17 134 ( n 5 ) + 6 29 114 ( n 5 ) + 24 67 33 ( n 5 ) + 21 25 2 ( n 5 ) + 206 194 385 + 4 678 11 7 + 6 41 7 + 24 29 + 2 110 19 + 3 43 14 + 3 53 19 + 4 138 77 + 4 94 55 + 4 786 737 + 4 78 79 + 4 74 77 + 2 82 95 + 6 190 287 + 24 22 41 7 + 6 65 133 + 60 3 7 7 + 12 10 41 + 6 115 779 + 12 10 77 + 12 34 287 + 12 2 17 + 12 78 779 + 24 5 91 + 60 2 247 + 36 2 553
Now we find G A 5 ( G 1 ) as
G A 5 ( G 1 ) = a b E ( G ) 2 S ( a ) S ( b ) ( S ( a ) + S ( b ) ) = j = 18 86 a b E j ( G ) 2 S ( a ) S ( b ) ( S ( a ) + S ( b ) ) .
Using the edge partition from Table 2, we get
G A 5 ( G 1 ) = 65 1066 | E 18 ( G 1 ) | + 5 91 | E 19 ( G 1 ) | + 5 494 | E 20 ( G 1 ) | + 75 1372 | E 21 ( G 1 ) | + 19 350 | E 22 ( G 1 ) | + 41 784 | E 23 ( G 1 ) | + 3 58 | E 24 ( G 1 ) | + 23 462 | E 25 ( G 1 ) | + 93 1876 | E 26 ( G 1 ) | + 105 2212 | E 27 ( G 1 ) | + 27 574 | E 28 ( G 1 ) | + 47 990 | E 29 ( G 1 ) | + 19 402 | E 30 ( G 1 ) | + 4 85 | E 31 ( G 1 ) | + 97 2070 | E 32 ( G 1 ) | + 14 315 | E 33 ( G 1 ) | + 43 896 | E 34 ( G 1 ) | + 11 232 | E 35 ( G 1 ) | + 53 1216 | E 36 ( G 1 ) | + 7 164 | E 37 ( G 1 ) | + 97 2178 | E 38 ( G 1 ) | + 98 2211 | E 39 ( G 1 ) | + 9 204 | E 40 ( G 1 ) | + 10 2277 | E 41 ( G 1 ) | + 110 2607 | E 56 ( G 1 ) | + 115 2772 | E 43 ( G 1 ) | + 55 1368 | E 44 ( G 1 ) | + 113 2844 | E 45 ( G 1 ) | + 29 738 | E 46 ( G 1 ) | + 59 1512 | E 47 ( G 1 ) | + 31 810 | E 48 ( G 1 ) | + 88 2009 | E 49 ( G 1 ) | + 95 2296 | E 50 ( G 1 ) | + 115 3116 | E 51 ( G 1 ) | + 97 2450 | E 52 ( G 1 ) | + 113 3234 | E 53 ( G 1 ) | + 126 3871 | E 54 ( G 1 ) | + 98 50 | E 55 ( G 1 ) | + 23 670 | E 56 ( G 1 ) | + 127 3950 | E 57 ( G 1 ) | + 7 203 | E 58 ( G 1 ) | + 5 154 | E 59 ( G 1 ) | + 65 2128 | E 60 ( G 1 ) | + 17 574 | E 61 ( G 1 ) | + 57 1682 | E 62 ( G 1 ) | + 31 952 | E 63 ( G 1 ) | + 69 2378 | E 64 ( G 1 ) | + 131 4422 | E 65 ( G 1 ) | + 33 1122 | E 66 ( G 1 ) | + 143 5214 | E 67 ( G 1 ) | + 37 1344 | E 68 ( G 1 ) | + 132 4489 | E 69 ( G 1 ) | + 134 4623 | E 70 ( G 1 ) | + 144 5293 | E 71 ( G 1 ) | + 149 5628 | E 72 ( G 1 ) | + 67 2312 | E 73 ( G 1 ) | + 135 4692 | E 74 ( G 1 ) | + 75 4856 | E 75 ( G 1 ) | + 136 4761 | E 76 ( G 1 ) | + 151 5796 | E 77 ( G 1 ) | + 159 6232 | E 78 ( G 1 ) | + 41 1710 | E 79 | ( G 1 ) + 161 6384 | E 80 ( G 1 ) | + 167 7110 | E 81 ( G 1 ) | + 81 6724 | E 82 ( G 1 ) | + 17 738 | E 83 ( G 1 ) | + 83 3528 | E 84 ( G 1 ) | + 86 3785 | E 85 ( G 1 ) | + 89 450 | E 86 ( G 1 ) | .
After simplification, we get
G A 5 ( G 1 ) = 40 13 14 ( n 4 ) 2 + 48 163 1659 ( n 2 ) + 12 7 10 n ( 3 n 2 15 n + 19 ) + 24 29 210 ( n 2 5 n + 6 ) + 16 13 77 ( n 2 5 n + 6 ) + 12 19 70 ( n 2 5 n + 6 ) + 18 5 21 ( n 2 5 n + 6 ) + 9 n 3 33 n 2 66 n + 3 14 2607 ( n 2 ) + 36 169 790 ( n 2 ) + 48 107 553 ( n 2 ) + 144 115 79 ( n 2 ) + 24 55 574 ( n 3 ) + 72 43 205 ( n 3 ) + 216 59 82 ( n 3 ) + 64 19 41 ( n 3 ) + 6 17 253 ( n 4 ) 2 + 8 11 230 ( n 4 ) 2 + 16 17 161 ( n 4 ) 2 + 12 73 5293 ( n 4 ) + 3 17 4623 ( n 4 ) + 6 25 2211 ( n 4 ) + 24 97 2010 ( n 4 ) + 48 151 1407 ( n 4 ) + 24 35 1189 ( n 4 ) + 48 137 1173 ( n 4 ) + 8 21 986 ( n 4 ) + 48 101 561 ( n 4 ) + 48 49 510 ( n 4 ) + 48 95 469 ( n 4 ) + 48 43 406 ( n 4 ) + 24 19 357 ( n 4 ) + 20 43 158 ( n 4 ) + 40 39 134 ( n 4 ) + 32 15 29 ( n 4 ) + 12 ( n 4 ) + 36 ( n 5 ) + 48 5214 145 + 48 4422 133 + 48 1558 79 + 48 1122 67 + 48 1066 67 + 16 779 39 + 96 574 97 + 16 494 17 + 48 462 47 + 32 287 23 + 16 266 11 + 96 231 61 + 32 203 19 + 72 190 83 + 48 154 25 + 96 91 41 + 21 79 16 + 336 66 115 + 3 55 + 28 41 15 + 32 38 9 + 36 19 7 + 192 7 11 + 1336 2 33 + 258  □

2.2. Results for Hex-Derived Cage Network (HDCN2(m,n))

In this portion, we find some degree-based topological indices for Hex-Derived Cage network ( H D C N 2 ( m , n ) ) . We calculate the general Randić index R α ( G ) with α = { 1 , 1 , 1 2 , 1 2 } , A B C , G A , A B C 4 and G A 5 in the the below theorems for ( H D C N 2 ( m , n ) ) .
Theorem 4.
Let G 2 H D C N 2 ( m , n ) be the Hex-Derived Cage network, then its general R a n d i c ´ index is equal to
R α ( G 2 ) = 6 ( 486 n 3 1068 n 2 + 312 n + 293 ) , α = 1 ; 6 ( 9 ( 2 2 + 3 ) n 3 + ( 2 30 + 2 15 + 3 6 + 6 5 + 12 3 60 2 81 ) n 2 + 2 ( 35 2 30 + 21 15 + 4 10 + 3 7 + 2 6 12 5 20 3 + 30 2 + 14 ) n + 2 42 4 35 + 4 30 4 21 16 10 12 7 20 6 + 24 5 + 48 3 12 2 + 39 ) , α = 1 2 ; 297675 n 3 445655 n 2 + 282283 n 40155 529200 , α = 1 ; 3 4 ( 2 2 + 3 ) n 3 + ( 4 5 + 2 3 5 2 + 2 6 5 + 2 3 + 3 5 83 12 ) n 2 + ( 461 105 ) + 6 2 5 + 3 7 3 5 2 3 4 6 5 + 5 2 5 3 16 5 + 4 7 + 12 35 ) n 24 35 8 7 + 16 5 + 8 3 2 + 4 6 5 + 2 6 7 2 2 3 2 3 7 12 2 5 + 13 14 , α = 1 2 .
Proof. 
Let G 2 be the Hex-Derived Cage network ( H D C N 2 ( m , n ) ) where m = n 5 . The edge set of H D C N 2 ( m , n ) is divided into twenty partitions based on the degree of end vertices. Table 3 shows these edge partition of H D C N 2 ( m , n ) .
R α ( G 2 ) = a b E ( G ) ( d ( a ) d ( b ) ) α
For α = 1
R 1 ( G 2 ) = j = 1 20 a b E j ( G ) d e g ( u ) · d e g ( v )
Using the edge partition from Table 3, we get
R 1 ( G 2 ) = 25 | E 1 ( G 2 ) | + 30 | E 2 ( G 2 ) | + 35 | E 3 ( G 2 ) | + 40 | E 4 ( G 2 ) | + 45 | E 5 ( G 2 ) | + 60 | E 6 ( G 2 ) | + 36 | E 7 ( G 2 ) | + 42 | E 8 ( G 2 ) | + 48 | E 9 ( G 2 ) | + 54 | E 10 ( G 2 ) | + 72 | E 11 ( G 2 ) | + 49 | E 12 ( G 2 ) | + 63 | E 13 ( G 2 ) | + 84 | E 14 ( G 2 ) | + 64 | E 15 ( G 2 ) | + 72 | E 16 ( G 2 ) | + 96 | E 17 ( G 2 ) | + 81 | E 18 ( G 2 ) | + 108 | E 19 ( G 2 ) | + 144 | E 20 ( G 2 ) |
After simplification, we get
R 1 ( G 2 ) = 6 ( 486 n 3 1068 n 2 + 312 n + 293 )
For α = 1 2
R 1 2 ( G 2 ) = j = 1 20 a b E j ( G ) d ( a ) · d ( b )
Using edge partition from Table 3, we get
R 1 2 ( G 2 ) = 5 | E 1 ( G 2 ) | + 30 | E 2 ( G 2 ) | + 35 | E 3 ( G 2 ) | + 2 10 | E 4 ( G 2 ) | + 3 5 | E 5 ( G 2 ) | + 2 15 | E 6 ( G 2 ) | + 6 | E 7 ( G 2 ) | + 42 | E 8 ( G 2 ) | + 4 3 | E 9 ( G 2 ) | + 3 6 | E 10 ( G 2 ) | + 6 2 | E 11 ( G 2 ) | + 7 | E 12 ( G 2 ) | + 3 7 | E 13 ( G 2 ) | + 2 21 | E 14 ( G 2 ) | + 8 | E 15 ( G 2 ) | + 6 2 | E 16 ( G 2 ) | + 4 6 | E 17 ( G 2 ) | + 9 | E 18 ( G 2 ) | + 6 3 | E 19 ( G 2 ) | + 12 | E 20 ( G 2 ) |
After simplification, we get
R 1 2 ( G 2 ) = 6 ( 9 ( 2 2 + 3 ) n 3 + ( 2 30 + 2 15 + 3 6 + 6 5 + 12 3 60 2 81 ) n 2 + 2 ( 35 2 30 + 21 15 + 4 10 + 3 7 + 2 6 12 5 20 3 + 30 2 + 14 ) n + 2 42 4 35 + 4 30 4 21 16 10 12 7 20 6 + 24 5 + 48 3 12 2 + 39 ) For α = 1
R 1 ( G 2 ) = j = 1 20 a b E j ( G ) 1 d ( a ) · d ( b )
R 1 ( G 2 ) = 1 25 | E 1 ( G 2 ) | + 1 30 | E 2 ( G 2 ) | + 1 35 | E 3 ( G 2 ) | + 1 40 | E 4 ( G 2 ) | + 1 45 | E 5 ( G 2 ) | + 1 60 | E 6 ( G 2 ) | + 1 36 | E 7 ( G 2 ) | + 1 42 | E 8 ( G 2 ) | + 1 48 | E 9 ( G 2 ) | + 1 54 | E 10 ( G 2 ) | + 1 72 | E 11 ( G 2 ) | + 1 49 | E 12 ( G 2 ) | + 1 63 | E 13 ( G 2 ) | + 1 84 | E 14 ( G 2 ) | + 1 64 | E 15 ( G 2 ) | + 1 72 | E 16 ( G 2 ) | + 1 96 | E 17 ( G 2 ) | + 1 81 | E 18 ( G 2 ) | + 1 108 | E 19 ( G 2 ) | + 1 144 | E 20 ( G 2 ) |
After simplification, we get
R 1 ( G 2 ) = 297675 n 3 445655 n 2 + 282283 n 40155 529200
For α = 1 2
R 1 2 ( G 2 ) = j = 1 20 a b E j ( G ) 1 d ( a ) · d ( b )
R 1 2 ( G 2 ) = 1 5 | E 1 ( G 2 ) | + 1 30 | E 2 ( G 2 ) | + 1 35 | E 3 ( G 2 ) | + 1 2 10 | E 4 ( G 2 ) | + 1 3 5 | E 5 ( G 2 ) | + 1 2 15 | E 6 ( G 2 ) | + 1 6 | E 7 ( G 2 ) | + 1 42 | E 8 ( G 2 ) | + 1 4 3 | E 9 ( G 2 ) | + 1 3 6 | E 10 ( G 2 ) | + 1 6 2 | E 11 ( G 2 ) | + 1 7 | E 12 ( G 2 ) | + 1 3 7 | E 13 ( G 2 ) | + 1 2 21 | E 14 ( G 2 ) | + 1 8 | E 15 ( G 2 ) | + 1 6 2 | E 16 ( G 2 ) | + 1 4 6 | E 17 ( G 2 ) | + 1 9 | E 18 ( G 2 ) | + 1 6 3 | E 19 ( G 2 ) | + 1 12 | E 20 ( G 2 ) |
After simplification, we get
R 1 2 ( G 2 ) = 3 4 ( 2 2 + 3 ) n 3 + ( 4 5 + 2 3 5 2 + 2 6 5 + 2 3 + 3 5 83 12 ) n 2 + ( 461 105 ) + 6 2 5 + 3 7 3 5 2 3 4 6 5 + 5 2 5 3 16 5 + 4 7 + 12 35 ) n 24 35 8 7 + 16 5 + 8 3 2 + 4 6 5 + 2 6 7 2 2 3 2 3 7 12 2 5 + 13 14  □
In this theorem, we find the first Zagreb index for hex-derived cage network G 2 .
Theorem 5.
For Hex-Derived Cage Network ( G 2 ), the first Zagreb index is equal to
M 1 ( G 2 ) = 12 ( 54 n 3 109 n 2 + 34 n + 21 )
Proof. 
Let G 2 be the Hex-Derived Cage Network ( G 2 ). Using the edge partition from Table 3, we have
M 1 ( G 2 ) = a b E ( G ) ( d ( a ) + d ( b ) ) = j = 1 20 a b E j ( G ) ( d ( a ) + d ( b ) )
M 1 ( G 2 ) = 10 | E 1 ( G 2 ) | + 11 | E 2 ( G 2 ) | + 12 | E 3 ( G 2 ) | + 13 | E 4 ( G 2 ) | + 14 | E 5 ( G 2 ) | + 17 | E 6 ( G 2 ) | + 12 | E 7 ( G 2 ) | + 13 | E 8 ( G 2 ) | + 14 | E 9 ( G 2 ) | + 15 | E 10 ( G 2 ) | + 18 | E 11 ( G 2 ) | + 14 | E 12 ( G 2 ) | + 16 | E 13 ( G 2 ) | + 19 | E 14 ( G 2 ) | + 16 | E 15 ( G 2 ) | + 17 | E 16 ( G 2 ) | + 20 | E 17 ( G 2 ) | + 18 | E 18 ( G 2 ) | + 21 | E 19 ( G 2 ) | + 24 | E 20 ( G 2 ) | .
After simplification, we get
M 1 ( G 2 ) = 12 ( 54 n 3 109 n 2 + 34 n + 21 )
 □
In below theorem, we calculate the A B C , G A , A B C 4 and G A 5 indices of Hex-Derived Cage Network G 2 .
Theorem 6.
Let G 2 be the Hex-Derived Cage Network for every positive integer m = n 5 ; then we have
  • A B C ( G 2 ) = 2 2 ( 3 n 3 10 n 2 + 8 n + 2 ) + 1 2 11 2 n ( 3 n 2 11 n + 10 ) + 5 2 n ( 3 n 2 11 n + 10 ) + 6 6 5 ( n 2 2 n + 2 ) + 16 3 ( n 2 5 n + 6 ) + 3 n ( n 1 ) + 12 2 n 5 + 6 n + 26 3 ( n 2 ) 2 + 2 19 3 ( n 2 ) 2 + 8 3 5 ( n 2 ) 2 + 30 ( n 2 ) + 51 7 ( n 2 ) + 6 22 5 ( n 2 ) + 6 3 ( n 2 ) + 4 2 ( n 2 ) + 12 2 7 ( n 2 ) + 3 7 2 ( n 3 ) + 12 7 3 ( n 3 ) + 2 66 7
  • G A ( G 2 ) = 4 2 ( 3 n 3 10 n 2 + 8 n + 2 ) + 24 11 30 ( n 2 2 n + 2 ) + 18 n 3 54 n 2 + 24 17 15 ( n 1 ) n + 48 3 n 7 + 12 n + 12 5 6 ( n 2 ) 2 + 36 7 5 ( n 2 ) 2 + 48 7 3 ( n 2 ) 2 + 2 35 ( n 2 ) + 24 19 21 ( n 2 ) + 96 13 10 ( n 2 ) + 9 2 7 ( n 2 ) + 48 5 6 ( n 2 ) + 144 17 2 ( n 2 ) + 12 ( n 3 ) + 24 42 13 + 54 .
  • A B C 4 ( G 2 ) = 1 18 107 2 n ( 3 n 2 17 n + 24 ) + 1 9 53 2 ( 3 n 3 13 n 2 + 16 n 10 ) + 4 9 5 n ( 3 n 2 15 n + 19 ) + 2 7 534 7 ( n 2 5 n + 6 ) + 3 102 101 ( n 2 5 n + 6 ) + 2 69 101 ( n 2 5 n + 6 ) + 24 7 37 101 ( n 2 5 n + 6 ) + 3 94 707 ( n 2 5 n + 6 ) + 60 101 2 ( n 2 6 n + 8 ) + 15 19 6 ( n 2 9 n + 20 ) + 7 2 95 ( 3 n 8 ) + 1 7 202 3 ( n 2 ) 2 + 4 7 258 13 ( n 2 ) + 4 13 19 ( n 2 ) + 5 3 2 ( n 2 ) + 12 7 142 95 ( n 2 ) + 67 95 ( n 2 ) + 12 194 9595 ( n 2 ) + 24 11 1235 ( n 2 ) + 129 5 ( n 3 ) + 2 302 33 ( n 3 ) + 4 3 205 33 ( n 3 ) + 274 55 ( n 3 ) + 2 145 33 ( n 3 ) + 3 7 123 19 ( n 4 ) 2 + 2 174 133 ( n 4 ) 2 + 30 7 1919 ( n 4 ) 2 + 4 35 366 ( n 4 ) + 2 5 447 19 ( n 4 ) + 6 5 206 13 ( n 4 ) + 2 3 14 ( n 4 ) + 4 46 35 ( n 4 ) + 2 42 37 ( n 4 ) + 12 5 46 65 ( n 4 ) + 24 5 58 101 ( n 4 ) + 3 37 65 ( n 4 ) + 6 2 13 ( n 4 ) + 6 38 259 ( n 4 ) + 6 2 19 ( n 4 ) + 6 334 3515 ( n 4 ) + 6 346 3737 ( n 4 ) + 66 7 2 37 ( n 4 ) + 72 2 715 ( n 4 ) + 56 33 ( n 4 ) + 6 37 146 ( n 5 ) + 8 25 37 ( n 5 ) + 96 65 2 ( n 5 ) + 32 37 + 2 249 37 + 8 6 5 + 2 202 35 + 2 109 21 + 8 106 35 3 + 66 23 + 183 2 37 + 4 678 511 + 2 70 53 + 48 30 73 7 + 8 14 37 + 8 330 949 + 8 134 511 + 8 6 23 + 12 127 851 + 6 3 23 + 32 10 77 3 + 12 290 2701 + 12 17 161 + 12 21 253 + 24 30 689 + 24 2 65 + 12 146 5035 + 24 166 6935 + 16 6 265 + 48 31 3869 + 48 43 7373 .
  • G A 5 ( G 2 ) = 3 n ( 3 n 2 17 n + 24 ) + 4 2 n ( 3 n 2 15 n + 19 ) + 12 143 4242 ( n 2 5 n + 6 ) + 28 25 101 ( n 2 5 n + 6 ) + 24 13 42 ( n 2 5 n + 6 ) + 2424 209 ( n 2 5 n + 6 ) + 36 149 570 ( 3 n 8 ) + 9 n 3 21 n 2 90 n + 252 103 6 ( n 2 ) 2 + 6 49 9595 ( n 2 ) + 12 67 3705 ( n 2 ) + 72 203 285 ( n 2 ) + 7 6 95 ( n 2 ) + 21 11 39 ( n 2 ) + 144 17 2 ( n 2 ) + 144 139 110 ( n 3 ) + 72 17 66 ( n 3 ) + 9120 33 ( n 3 ) 1127 + 48 11 30 ( n 3 ) + 16 59 1919 ( n 4 ) 2 + 24 59 798 ( n 4 ) 2 + 168 125 19 ( n 4 ) 2 + 24 175 7474 ( n 4 ) + 24 169 7030 ( n 4 ) + 24 113 2886 ( n 4 ) + 8 25 1406 ( n 4 ) + 12 29 777 ( n 4 ) + 36 41 715 ( n 4 ) + 15 11 303 ( n 4 ) + 1350 791 195 ( n 4 ) + 9 8 111 ( n 4 ) + 56 41 74 ( n 4 ) + 240 151 57 ( n 4 ) + 1496 217 26 ( n 4 ) + 80 13 14 ( n 4 ) + 210 31 3 ( n 4 ) + 12 ( n 4 ) + 36 ( n 5 ) + 8 7373 29 + 2 6935 7 + 16 5402 49 + 6 5035 37 + 8 3869 21 + 48 3066 115 + 3 2847 7 + 12 2067 23 + 32 851 43 + 18 511 17 + 9 455 8 + 72 318 107 + 36 265 49 + 36 259 25 + 288 253 191 + 60 219 37 + 72 185 41 + 288 161 155 + 144 138 73 + 144 115 137 + 192 111 85 + 8 77 3 + 168 73 61 + 288 70 103 + 732 69 175 + 4 35 + 192 21 37 + 258 .
Proof. 
Using the edge partition from Table 3, we find A B C as
A B C ( G 2 ) = a b E ( G ) d ( a ) + d ( b ) 2 d ( a ) · d ( b ) = j = 1 20 a b E j ( G ) d ( a ) + d ( b ) 2 d ( a ) · d ( b )
A B C ( G 2 ) = 2 2 5 | E 1 ( G 2 ) | + 3 10 | E 2 ( G 2 ) | + 2 7 | E 3 ( G 2 ) | + 11 2 10 | E 4 ( G 2 ) | + 2 3 3 5 | E 5 ( G 2 ) | + 1 2 | E 6 ( G 2 ) | + 5 18 | E 7 ( G 2 ) | + 11 42 | E 8 ( G 2 ) | + 1 2 | E 9 ( G 2 ) | + 13 54 | E 10 ( G 2 ) | + 2 3 | E 11 ( G 2 ) | + 2 3 7 | E 12 ( G 2 ) | + 14 63 | E 13 ( G 2 ) | + 17 84 | E 14 ( G 2 ) | + 7 32 | E 15 ( G 2 ) | + 5 24 | E 16 ( G 2 ) | + 3 16 | E 17 ( G 2 ) | + 16 81 | E 18 ( G 2 ) | + 19 108 | E 19 ( G 2 ) | + 11 72 | E 20 ( G 2 ) | .
After simplification, we get
A B C ( G 2 ) = 2 2 ( 3 n 3 10 n 2 + 8 n + 2 ) + 1 2 11 2 n ( 3 n 2 11 n + 10 ) + 5 2 n ( 3 n 2 11 n + 10 ) + 6 6 5 ( n 2 2 n + 2 ) + 16 3 ( n 2 5 n + 6 ) + 3 n ( n 1 ) + 12 2 n 5 + 6 n + 26 3 ( n 2 ) 2 + 2 19 3 ( n 2 ) 2 + 8 3 5 ( n 2 ) 2 + 30 ( n 2 ) + 51 7 ( n 2 ) + 6 22 5 ( n 2 ) + 6 3 ( n 2 ) + 4 2 ( n 2 ) + 12 2 7 ( n 2 ) + 3 7 2 ( n 3 ) + 12 7 3 ( n 3 ) + 2 66 7
Using the edge partition from Table 3, we find G A as
G A ( G 2 ) = 1 | E 1 ( G 2 ) | + 2 30 11 | E 2 ( G 2 ) | + 35 6 | E 3 ( G 2 ) | + 4 10 13 | E 4 ( G 2 ) | + 3 5 7 | E 5 ( G 2 ) | + 4 15 17 | E 6 ( G 2 ) | + 1 | E 7 ( G 2 ) | + 2 42 13 | E 8 ( G 2 ) | + 2 12 7 | E 9 ( G 2 ) | + 2 54 15 | E 10 ( G 2 ) | + 2 2 3 | E 11 ( G 2 ) | + 1 | E 12 ( G 2 ) | + 3 7 8 | E 13 ( G 2 ) | + 4 21 19 | E 14 ( G 2 ) | + 1 | E 15 ( G 2 ) | + 12 2 17 | E 16 ( G 2 ) | + 23 5 | E 17 ( G 2 ) | + 1 | E 18 ( G 2 ) | + 2 3 7 | E 19 ( G 2 ) | + 1 | E 20 ( G 2 ) | .
After simplification, we get
G A ( G 2 ) = 4 2 ( 3 n 3 10 n 2 + 8 n + 2 ) + 24 11 30 ( n 2 2 n + 2 ) + 18 n 3 54 n 2 + 24 17 15 ( n 1 ) n + 48 3 n 7 + 12 n + 12 5 6 ( n 2 ) 2 + 36 7 5 ( n 2 ) 2 + 48 7 3 ( n 2 ) 2 + 2 35 ( n 2 ) + 24 19 21 ( n 2 ) + 96 13 10 ( n 2 ) + 9 2 7 ( n 2 ) + 48 5 6 ( n 2 ) + 144 17 2 ( n 2 ) + 12 ( n 3 ) + 24 42 13 + 54 .
If we suppose an edge partition based on degree sum of neighbors of end vertices, then the edge set E ( G 2 ) can be divided into seventy six edge partition E j ( G 2 ) , 21 j 96 . Table 4 shows these edge partitions.
From Equation (7), we get
A B C 4 ( G 2 ) = a b E ( G ) S ( a ) + S ( b ) 2 S ( a ) S ( b ) = j = 21 96 a b E j ( G ) S ( a ) + S ( b ) 2 S ( a ) S ( b ) .
Using the edge partition from Table 3, we get
A B C 4 ( G 2 ) = 72 1369 | E 21 ( G 2 ) | + 4 333 | E 22 ( G 2 ) | + 83 1776 | E 23 ( G 2 ) | + 98 2321 | E 24 ( G 2 ) | + 127 3404 | E 25 ( G 2 ) | + 74 1443 | E 26 ( G 2 ) | + 86 1911 | E 27 ( G 2 ) | + 90 2067 | E 28 ( G 2 ) | + 42 2106 | E 29 ( G 2 ) | + 110 2847 | E 30 ( G 2 ) | + 111 2886 | E 31 ( G 2 ) | + 44 1235 | E 32 ( G 2 ) | + 43 940 | E 33 ( G 2 ) | + 101 2520 | E 34 ( G 2 ) | + 103 2600 | E 35 ( G 2 ) | + 137 3960 | E 36 ( G 2 ) | + 89 2058 | E 37 ( G 2 ) | + 113 3066 | E 38 ( G 2 ) | + 17 1554 | E 39 ( G 2 ) | + 115 3150 | E 40 ( G 2 ) | + 29 798 | E 41 ( G 2 ) | + 141 4242 | E 42 ( G 2 ) | + 96 2385 | E 43 ( G 2 ) | + 106 2835 | E 44 ( G 2 ) | + 27 558 | E 45 ( G 2 ) | + 5 648 | E 46 ( G 2 ) | + 109 3024 | E 47 ( G 2 ) | + 111 3120 | E 48 ( G 2 ) | + 69 228 | E 49 ( G 2 ) | + 145 4752 | E 50 ( G 2 ) | + 101 2646 | E 51 ( G 2 ) | + 120 3577 | E 52 ( G 2 ) | + 11 3626 | E 53 ( G 2 ) | + 122 3675 | E 54 ( G 2 ) | + 123 3724 | E 55 ( G 2 ) | + 142 4655 | E 56 ( G 2 ) | + 148 4949 | E 57 ( G 2 ) | + 105 2862 | E 58 ( G 2 ) | + 124 3869 | E 59 ( G 2 ) | + 146 5035 | E 60 ( G 2 ) | + 53 1458 | E 61 ( G 2 ) | + 63 1998 | E 62 ( G 2 ) | + 105 621 | E 63 ( G 2 ) | + 147 5130 | E 64 ( G 2 ) | + 151 5346 | E 65 ( G 2 ) | + 153 5454 | E 66 ( G 2 ) | + 20 729 | E 67 ( G 2 ) | + 126 4095 | E 68 ( G 2 ) | + 134 4599 | E 69 ( G 2 ) | + 153 5796 | E 70 ( G 2 ) | + 160 6237 | E 71 ( G 2 ) | + 128 4225 | E 72 ( G 2 ) | + 46 1625 | E 73 ( G 2 ) | + 54 2145 | E 74 ( G 2 ) | + 145 5402 | E 75 ( G 2 ) | + 116 5475 | E 76 ( G 2 ) | + 166 6935 | E 77 ( G 2 ) | + 172 7373 | E 78 ( G 2 ) | + 73 2738 | E 79 ( G 2 ) | + 37 1406 | E 80 ( G 2 ) | + 167 7030 | E 81 ( G 2 ) | + 173 7474 | E 82 ( G 2 ) | + 148 5625 | E 83 ( G 2 ) | + 149 5700 | E 84 ( G 2 ) | + 58 2525 | E 85 ( G 2 ) | + 75 2888 | E 86 ( G 2 ) | + 175 7676 | E 87 ( G 2 ) | + 189 9108 | E 88 ( G 2 ) | + 99 4948 | E 89 ( G 2 ) | + 194 9595 | E 90 ( G 2 ) | + 201 10260 | E 91 ( G 2 ) | + 14 9801 | E 92 ( G 2 ) | + 205 10692 | E 93 ( G 2 ) | + 200 10201 | E 94 ( G 2 ) | + 207 10908 | E 95 ( G 2 ) | + 3 162 | E 96 ( G 2 ) | .
After simplification, we have
A B C 4 ( G 2 ) = 1 18 107 2 n ( 3 n 2 17 n + 24 ) + 1 9 53 2 ( 3 n 3 13 n 2 + 16 n 10 ) + 4 9 5 n ( 3 n 2 15 n + 19 ) + 2 7 534 7 ( n 2 5 n + 6 ) + 3 102 101 ( n 2 5 n + 6 ) + 2 69 101 ( n 2 5 n + 6 ) + 24 7 37 101 ( n 2 5 n + 6 ) + 3 94 707 ( n 2 5 n + 6 ) + 60 101 2 ( n 2 6 n + 8 ) + 15 19 6 ( n 2 9 n + 20 ) + 7 2 95 ( 3 n 8 ) + 1 7 202 3 ( n 2 ) 2 + 4 7 258 13 ( n 2 ) + 4 13 19 ( n 2 ) + 5 3 2 ( n 2 ) + 12 7 142 95 ( n 2 ) + 67 95 ( n 2 ) + 12 194 9595 ( n 2 ) + 24 11 1235 ( n 2 ) + 129 5 ( n 3 ) + 2 302 33 ( n 3 ) + 4 3 205 33 ( n 3 ) + 274 55 ( n 3 ) + 2 145 33 ( n 3 ) + 3 7 123 19 ( n 4 ) 2 + 2 174 133 ( n 4 ) 2 + 30 7 1919 ( n 4 ) 2 + 4 35 366 ( n 4 ) + 2 5 447 19 ( n 4 ) + 6 5 206 13 ( n 4 ) + 2 3 14 ( n 4 ) + 4 46 35 ( n 4 ) + 2 42 37 ( n 4 ) + 12 5 46 65 ( n 4 ) + 24 5 58 101 ( n 4 ) + 3 37 65 ( n 4 ) + 6 2 13 ( n 4 ) + 6 38 259 ( n 4 ) + 6 2 19 ( n 4 ) + 6 334 3515 ( n 4 ) + 6 346 3737 ( n 4 ) + 66 7 2 37 ( n 4 ) + 72 2 715 ( n 4 ) + 56 33 ( n 4 ) + 6 37 146 ( n 5 ) + 8 25 37 ( n 5 ) + 96 65 2 ( n 5 ) + 32 37 + 2 249 37 + 8 6 5 + 2 202 35 + 2 109 21 + 8 106 35 3 + 66 23 + 183 2 37 + 4 678 511 + 2 70 53 + 48 30 73 7 + 8 14 37 + 8 330 949 + 8 134 511 + 8 6 23 + 12 127 851 + 6 3 23 + 32 10 77 3 + 12 290 2701 + 12 17 161 + 12 21 253 + 24 30 689 + 24 2 65 + 12 146 5035 + 24 166 6935 + 16 6 265 + 48 31 3869 + 48 43 7373
From Equation (8), we get
G A 5 ( G 2 ) = a b E ( G ) 2 S ( a ) S ( b ) ( S ( a ) + S ( b ) ) = j = 21 96 a b E j ( G ) 2 S ( a ) S ( b ) ( S ( a ) S ( b ) ) .
Using the edge partition from Table 4, we get
G A 5 ( G 2 ) = 1 | E 21 ( G 2 ) | + 1665 41 | E 22 ( G 2 ) | + 2 1776 85 | E 23 ( G 2 ) | + 2331 50 | E 24 ( G 2 ) | + 2 3404 129 | E 25 ( G 2 ) | + 1443 38 | E 26 ( G 2 ) | + 1911 44 | E 27 ( G 2 ) | + 2067 46 | E 28 ( G 2 ) | + 2 2106 93 | E 29 ( G 2 ) | + 2847 56 | E 30 ( G 2 ) | + 2 2886 113 | E 31 ( G 2 ) | + 3705 67 | E 32 ( G 2 ) | + 1920 44 | E 33 ( G 2 ) | + 2 2520 103 | E 34 ( G 2 ) | + 2 2600 105 | E 35 ( G 2 ) | + 2 3960 139 | E 36 ( G 2 ) | + 2 2058 91 | E 37 ( G 2 ) | + 2 3066 115 | E 38 ( G 2 ) | + 3108 58 | E 39 ( G 2 ) | + 2 3150 117 | E 40 ( G 2 ) | + 3192 59 | E 41 ( G 2 ) | + 2 4242 143 | E 42 ( G 2 ) | + 2385 49 | E 43 ( G 2 ) | + 2835 54 | E 44 ( G 2 ) | + 2 4140 137 | E 45 ( G 2 ) | + 2592 51 | E 46 ( G 2 ) | + 2 3024 111 | E 47 ( G 2 ) | + 2 3120 113 | E 48 ( G 2 ) | + 4416 70 | E 49 ( G 2 ) | + 2 4752 147 | E 50 ( G 2 ) | + 2 2646 103 | E 51 ( G 2 ) | + 3577 61 | E 52 ( G 2 ) | + 2 3626 123 | E 53 ( G 2 ) | + 3675 62 | E 54 ( G 2 ) | + 2 3724 125 | E 55 ( G 2 ) | + 4655 72 | E 56 ( G 2 ) | + 4949 75 | E 57 ( G 2 ) | + 2 2862 107 | E 58 ( G 2 ) | + 3869 63 | E 59 ( G 2 ) | + 5035 74 | E 60 ( G 2 ) | + 1 | E 61 ( G 2 ) | + 3996 64 | E 62 ( G 2 ) | + 4968 73 | E 63 ( G 2 ) | + 2 5130 149 | E 64 ( G 2 ) | + 2 5346 153 | E 65 ( G 2 ) | + 2 5454 155 | E 66 ( G 2 ) | + 5832 81 | E 67 ( G 2 ) | + 4095 64 | E 68 ( G 2 ) | + 4599 68 | E 69 ( G 2 ) | + 2 5796 155 | E 70 ( G 2 ) | + 6237 81 | E 71 ( G 2 ) | + 1 | E 72 ( G 2 ) | + 4875 70 | E 73 ( G 2 ) | + 6435 82 | E 74 ( G 2 ) | + 2 5402 147 | E 75 ( G 2 ) | + 5475 74 | E 76 ( G 2 ) | + 6935 84 | E 77 ( G 2 ) | + 7373 87 | E 78 ( G 2 ) | + 1 | E 79 ( G 2 ) | + 5624 75 | E 80 ( G 2 ) | + 2 7030 169 | E 81 ( G 2 ) | + 2 7474 175 | E 82 ( G 2 ) | + 1 | E 83 ( G 2 ) | + 2 5700 151 | E 84 ( G 2 ) | + 7575 88 | E 85 ( G 2 ) | + 1 | E 86 ( G 2 ) | + 2 7676 177 | E 87 ( G 2 ) | + 2 9108 191 | E 88 ( G 2 ) | + 9936 100 | E 89 ( G 2 ) | + 9595 98 | E 90 ( G 2 ) | + 2 10260 203 | E 91 ( G 2 ) | + 1 | E 92 ( G 2 ) | + 2 10692 207 | E 93 ( G 2 ) | + 1 | E 94 ( G 2 ) | + 2 10908 209 | E 95 ( G 2 ) | + 1 | E 96 ( G 2 ) | .
After simplification, we get
G A 5 ( G 2 ) = 3 n ( 3 n 2 17 n + 24 ) + 4 2 n ( 3 n 2 15 n + 19 ) + 12 143 4242 ( n 2 5 n + 6 ) + 28 25 101 ( n 2 5 n + 6 ) + 24 13 42 ( n 2 5 n + 6 ) + 2424 209 ( n 2 5 n + 6 ) + 36 149 570 ( 3 n 8 ) + 9 n 3 21 n 2 90 n + 252 103 6 ( n 2 ) 2 + 6 49 9595 ( n 2 ) + 12 67 3705 ( n 2 ) + 72 203 285 ( n 2 ) + 7 6 95 ( n 2 ) + 21 11 39 ( n 2 ) + 144 17 2 ( n 2 ) + 144 139 110 ( n 3 ) + 72 17 66 ( n 3 ) + 9120 33 ( n 3 ) 1127 + 48 11 30 ( n 3 ) + 16 59 1919 ( n 4 ) 2 + 24 59 798 ( n 4 ) 2 + 168 125 19 ( n 4 ) 2 + 24 175 7474 ( n 4 ) + 24 169 7030 ( n 4 ) + 24 113 2886 ( n 4 ) + 8 25 1406 ( n 4 ) + 12 29 777 ( n 4 ) + 36 41 715 ( n 4 ) + 15 11 303 ( n 4 ) + 1350 791 195 ( n 4 ) + 9 8 111 ( n 4 ) + 56 41 74 ( n 4 ) + 240 151 57 ( n 4 ) + 1496 217 26 ( n 4 ) + 80 13 14 ( n 4 ) + 210 31 3 ( n 4 ) + 12 ( n 4 ) + 36 ( n 5 ) + 8 7373 29 + 2 6935 7 + 16 5402 49 + 6 5035 37 + 8 3869 21 + 48 3066 115 + 3 2847 7 + 12 2067 23 + 32 851 43 + 18 511 17 + 9 455 8 + 72 318 107 + 36 265 49 + 36 259 25 + 288 253 191 + 60 219 37 + 72 185 41 + 288 161 155 + 144 138 73 + 144 115 137 + 192 111 85 + 8 77 3 + 168 73 61 + 288 70 103 + 732 69 175 + 4 35 + 192 21 37 + 258 . □
The Comparison graphs for ABC, GA, A B C 4 and G A 5 in case of a Hex Derived Cage networks H D C N 1 ( m , n ) and H D C N 2 ( m , n ) of dimension m and n are shown in Figure 3 and Figure 4 respectively.

3. Conclusions

In this paper, certain degree-based topological indices, namely the general Randić index, atomic-bond connectivity index (ABC), geometric-arithmetic index (GA) and first Zagreb index were studied for the first time and analytical closed formulas for H D C N 1 ( m , n ) and H D C N 2 ( m , n ) cage networks were determined which will help the people working in network science to understand and explore the underlying topologies of these networks.
For the future, we are interested in designing some new architectures/networks and then study their topological indices which will be quite helpful to understand their underlying topologies.

Author Contributions

Data curation, U.M.; Funding acquisition, J.-B.L.; Methodology, J.-B.L.; Software, U.M.; Supervision, M.K.S.; Writing—original draft, H.A.

Funding

The work was partially supported by China Postdoctoral Science Foundation under grant No. 2017M621579 and Postdoctoral Science Foundation of Jiangsu Province under grant No. 1701081B, Project of Anhui Jianzhu University under Grant no. 2016QD116 and 2017dc03, Anhui Province Key Laboratory of Intelligent Building & Building Energy Saving.

Acknowledgments

The authors would like to thank all the respected reviewers for their suggestions and useful comments, which resulted in an improved version of this paper.

Conflicts of Interest

The authors declare no conflict of interest.

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Figure 1. Hex-Derived Network ( H D C N 1 ( 3 , n ) ).
Figure 1. Hex-Derived Network ( H D C N 1 ( 3 , n ) ).
Symmetry 10 00619 g001
Figure 2. Hex-Derived Network ( H D C N 2 ( 3 , n ) ).
Figure 2. Hex-Derived Network ( H D C N 2 ( 3 , n ) ).
Symmetry 10 00619 g002
Figure 3. Comparison of ABC, GA, A B C 4 and G A 5 for H D C N 1 ( m , n ) .
Figure 3. Comparison of ABC, GA, A B C 4 and G A 5 for H D C N 1 ( m , n ) .
Symmetry 10 00619 g003
Figure 4. Comparison of ABC, GA, A B C 4 and G A 5 for H D C N 2 ( m , n ) .
Figure 4. Comparison of ABC, GA, A B C 4 and G A 5 for H D C N 2 ( m , n ) .
Symmetry 10 00619 g004
Table 1. Edge partition of Hex-Derived Cage network ( H D C N 1 ) based on degrees of end vertices of each edge.
Table 1. Edge partition of Hex-Derived Cage network ( H D C N 1 ) based on degrees of end vertices of each edge.
( d u , d v ) where ab E ( G 1 ) Number of Edges ( d u , d v ) where ab E ( G 1 ) Number of Edges
E 1 = ( 3 , 6 ) 24 E 10 = ( 7 , 7 ) 6 n 18
E 2 = ( 3 , 7 ) 2 ( 6 n 12 ) E 11 = ( 7 , 9 ) 2 ( 6 n 12 )
E 3 = ( 3 , 8 ) 6 ( 6 n 12 ) E 12 = ( 7 , 12 ) 6 n 12
E 4 = ( 3 , 9 ) 18 n 2 72 n + 72 E 13 = ( 8 , 8 ) 2 ( 6 n 18 )
E 5 = ( 3 , 12 ) 18 n 3 54 n 2 + 42 n E 14 = ( 8 , 9 ) 2 ( 6 n 12 )
E 6 = ( 6 , 7 ) 12 E 15 = ( 8 , 12 ) 4 ( 6 n 12 )
E 7 = ( 6 , 8 ) 24 E 16 = ( 9 , 9 ) 12 n 2 60 n + 72
E 8 = ( 6 , 12 ) 12 E 17 = ( 9 , 12 ) 12 n 2 48 n + 48
E 9 = ( 12 , 12 ) 9 n 3 33 n 2 + 30 n
Table 2. Edge partition of Hex-Derived Cage network ( H D C N 1 ) based on degrees of end vertices of each edge.
Table 2. Edge partition of Hex-Derived Cage network ( H D C N 1 ) based on degrees of end vertices of each edge.
( S u , S v ) where ab E ( G 1 ) Number of Edges ( S u , S v ) where ab E ( G 1 ) Number of Edges
E 18 = ( 26 , 41 ) 24 E 53 = ( 49 , 66 ) 24
E 19 = ( 26 , 56 ) 24 E 54 = ( 49 , 79 ) 12
E 20 = ( 26 , 76 ) 24 E 55 = ( 50 , 50 ) 6 n 30
E 21 = ( 28 , 49 ) 24 E 56 = ( 50 , 67 ) 2 ( 6 n 24 )
E 22 = ( 28 , 50 ) 2 ( 6 n 24 ) E 57 = ( 50 , 79 ) 6 n 24
E 23 = ( 28 , 56 ) 24 E 58 = ( 56 , 58 ) 24
E 24 = ( 28 , 58 ) 4 ( 6 n 24 ) E 59 = ( 56 , 66 ) 24
E 25 = ( 28 , 66 ) 24 E 60 = ( 56 , 76 ) 24
E 26 = ( 28 , 67 ) 2 ( 6 n 24 ) E 61 = ( 56 , 82 ) 24
E 27 = ( 28 , 79 ) 2 ( 6 n 12 ) E 62 = ( 58 , 58 ) 2 ( 6 n 30 )
E 28 = ( 28 , 82 ) 2 ( 6 n 18 ) E 63 = ( 58 , 68 ) 2 ( 6 n 24 )
E 29 = ( 30 , 66 ) 24 E 64 = ( 58 , 82 ) 4 ( 6 n 24 )
E 30 = ( 30 , 67 ) 2 ( 6 n 24 ) E 65 = ( 66 , 67 ) 24
E 31 = ( 30 , 68 ) 4 ( 6 n 24 ) E 66 = ( 66 , 68 ) 24
E 32 = ( 30 , 69 ) 12 n 2 96 n + 192 E 67 = ( 66 , 79 ) 24
E 33 = ( 30 , 84 ) 6 n 2 30 n + 36 E 68 = ( 66 , 84 ) 24
E 34 = ( 32 , 56 ) 24 E 69 = ( 67 , 67 ) 2 ( 6 n 30 )
E 35 = ( 32 , 58 ) 2 ( 6 n 24 ) E 70 = ( 67 , 69 ) 2 ( 6 n 24 )
E 36 = ( 32 , 76 ) 24 E 71 = ( 67 , 79 ) 2 ( 6 n 24 )
E 37 = ( 32 , 82 ) 4 ( 6 n 18 ) E 72 = ( 67 , 84 ) 2 ( 6 n 24 )
E 38 = ( 33 , 66 ) 24 E 73 = ( 68 , 68 ) 2 ( 6 n 30 )
E 39 = ( 33 , 67 ) 2 ( 6 n 24 ) E 74 = ( 68 , 69 ) 2 ( 6 n 24 )
E 40 = ( 33 , 68 ) 2 ( 6 n 24 ) E 75 = ( 68 , 84 ) 4 ( 6 n 24 )
E 41 = ( 33 , 69 ) 6 n 2 48 n + 96 E 76 = ( 69 , 69 ) 12 n 2 108 n + 240
E 42 = ( 33 , 79 ) 2 ( 6 n 12 ) E 77 = ( 69 , 84 ) 12 n 2 96 n + 192
E 43 = ( 33 , 84 ) 12 n 2 60 n + 72 E 78 = ( 76 , 82 ) 24
E 44 = ( 36 , 76 ) 24 E 79 = ( 76 , 90 ) 12
E 45 = ( 36 , 79 ) 2 ( 6 n 12 ) E 80 = ( 79 , 84 ) 2 ( 6 n 12 )
E 46 = ( 36 , 82 ) 6 ( 6 n 18 ) E 81 = ( 79 , 90 ) 6 n 12
E 47 = ( 36 , 84 ) 18 n 2 90 n + 108 E 82 = ( 82 , 82 ) 2 ( 6 n 24 )
E 48 = ( 36 , 90 ) 18 n 3 90 n 2 + 114 n E 83 = ( 82 , 90 ) 4 ( 6 n 18 )
E 49 = ( 41 , 49 ) 12 E 84 = ( 84 , 84 ) 6 n 2 36 n + 48
E 50 = ( 41 , 56 ) 24 E 85 = ( 84 , 90 ) 12 n 2 60 n + 72
E 51 = ( 41 , 76 ) 12 E 86 = ( 90 , 90 ) 9 n 3 51 n 2 + 72 n
E 52 = ( 49 , 50 ) 12
Table 3. Edge partition of Hex-Derived Cage network ( H D C N 2 ) based on degrees of end vertices of each edge.
Table 3. Edge partition of Hex-Derived Cage network ( H D C N 2 ) based on degrees of end vertices of each edge.
( d u , d v ) where ab E ( G 2 ) Number of Edges ( d u , d v ) where ab E ( G 2 ) Number of Edges
E 1 = ( 5 , 5 ) 6 n E 11 = ( 6 , 12 ) 18 n 3 60 n 2 + 48 n + 12
E 2 = ( 5 , 6 ) 12 n 2 24 n + 24 E 12 = ( 7 , 7 ) 6 n 18
E 3 = ( 5 , 7 ) 2 ( 6 n 12 ) E 13 = ( 7 , 9 ) 2 ( 6 n 12 )
E 4 = ( 5 , 8 ) 4 ( 6 n 12 ) E 14 = ( 7 , 12 ) 6 n 12
E 5 = ( 5 , 9 ) 12 n 2 48 n + 48 E 15 = ( 8 , 8 ) 2 ( 6 n 18 )
E 6 = ( 5 , 12 ) 6 n 2 6 n E 16 = ( 8 , 9 ) 2 ( 6 n 12 )
E 7 = ( 6 , 6 ) 9 n 3 33 n 2 + 30 n E 17 = ( 8 , 12 ) 4 ( 6 n 12 )
E 8 = ( 6 , 7 ) 12 E 18 = ( 9 , 9 ) 12 n 2 60 n + 72
E 9 = ( 6 , 8 ) 12 n E 19 = ( 9 , 12 ) 12 n 2 48 n + 48
E 10 = ( 6 , 9 ) 6 n 2 24 n + 24 E 20 = ( 12 , 12 ) 9 n 3 33 n 2 + 30 n
Table 4. Edge partition of Hex-Derived Cage network ( H D C N 2 ) based on sum of degrees of end vertices of each edge.
Table 4. Edge partition of Hex-Derived Cage network ( H D C N 2 ) based on sum of degrees of end vertices of each edge.
( S u , S v ) where ab E ( G 2 ) Number of Edges ( S u , S v ) where ab E ( G 2 ) Number of Edges
E 21 = ( 37 , 37 ) 12 E 59 = ( 53 , 73 ) 24
E 22 = ( 37 , 45 ) 24 E 60 = ( 53 , 95 ) 12
E 23 = ( 37 , 48 ) 24 E 61 = ( 54 , 54 ) 9 n 3 39 n 2 + 48 n 30
E 24 = ( 37 , 63 ) 24 E 62 = ( 54 , 74 ) 2 ( 6 n 24 )
E 25 = ( 37 , 92 ) 24 E 63 = ( 54 , 92 ) 24
E 26 = ( 37 , 39 ) 6 n 12 E 64 = ( 54 , 95 ) 3 ( 6 n 16 )
E 27 = ( 39 , 49 ) 2 ( 6 n 12 ) E 65 = ( 54 , 99 ) 6 ( 6 n 18 )
E 28 = ( 39 , 53 ) 24 E 66 = ( 54 , 101 ) 18 n 2 90 n + 108
E 29 = ( 39 , 54 ) 2 ( 6 n 24 ) E 67 = ( 54 , 108 ) 18 n 3 90 n 2 + 114 n
E 30 = ( 39 , 73 ) 24 E 68 = ( 63 , 65 ) 24
E 31 = ( 39 , 74 ) 2 ( 6 n 24 ) E 69 = ( 63 , 73 ) 24
E 32 = ( 39 , 95 ) 2 ( 6 n 12 ) E 70 = ( 63 , 92 ) 24
E 33 = ( 40 , 48 ) 4 ( 6 n 18 ) E 71 = ( 63 , 99 ) 24
E 34 = ( 40 , 63 ) 24 E 72 = ( 65 , 65 ) 2 ( 6 n 30 )
E 35 = ( 40 , 65 ) 4 ( 6 n 24 ) E 73 = ( 65 , 75 ) 2 ( 6 n 24 )
E 36 = ( 40 , 99 ) 2 ( 6 n 18 ) E 74 = ( 65 , 99 ) 4 ( 6 n 24 )
E 37 = ( 42 , 49 ) 12 n 2 60 n + 72 E 75 = ( 73 , 74 ) 24
E 38 = ( 42 , 73 ) 24 E 76 = ( 73 , 75 ) 24
E 39 = ( 42 , 74 ) 2 ( 6 n 24 ) E 77 = ( 73 , 95 ) 24
E 40 = ( 42 , 75 ) 4 ( 6 n 24 ) E 78 = ( 73 , 101 ) 24
E 41 = ( 42 , 76 ) 12 n 2 96 n + 192 E 79 = ( 74 , 74 ) 2 ( 6 n 30 )
E 42 = ( 42 , 101 ) 6 n 2 30 n + 36 E 80 = ( 74 , 76 ) 2 ( 6 n 24 )
E 43 = ( 45 , 53 ) 12 E 81 = ( 74 , 95 ) 2 ( 6 n 24 )
E 44 = ( 45 , 63 ) 24 E 82 = ( 74 , 101 ) 2 ( 6 n 24 )
E 45 = ( 45 , 92 ) 12 E 83 = ( 75 , 75 ) 2 ( 6 n 30 )
E 46 = ( 48 , 54 ) 2 ( 6 n 12 ) E 84 = ( 75 , 76 ) 2 ( 6 n 24 )
E 47 = ( 48 , 63 ) 24 E 85 = ( 75 , 101 ) 4 ( 6 n 24 )
E 48 = ( 48 , 65 ) 2 ( 6 n 24 ) E 86 = ( 76 , 76 ) 12 n 2 108 n + 240
E 49 = ( 48 , 92 ) 24 E 87 = ( 76 , 101 ) 12 n 2 96 n + 192
E 50 = ( 48 , 99 ) 4 ( 6 n 18 ) E 88 = ( 92 , 99 ) 24
E 51 = ( 49 , 54 ) 6 n 2 24 n + 24 E 89 = ( 92 , 108 ) 12
E 52 = ( 49 , 73 ) 24 E 90 = ( 95 , 101 ) 2 ( 6 n 12 )
E 53 = ( 49 , 74 ) 2 ( 6 n 24 ) E 91 = ( 95 , 108 ) 6 n 12
E 54 = ( 49 , 75 ) 2 ( 6 n 24 ) E 92 = ( 99 , 99 ) 2 ( 6 n 24 )
E 55 = ( 49 , 76 ) 6 n 2 48 n + 96 E 93 = ( 99 , 108 ) 4 ( 6 n 18 )
E 56 = ( 49 , 95 ) 2 ( 6 n 12 ) E 94 = ( 101 , 101 ) 6 n 2 36 n + 48
E 57 = ( 49 , 101 ) 12 n 2 60 n + 72 E 95 = ( 101 , 108 ) 12 n 2 60 n + 72
E 58 = ( 53 , 54 ) 12 E 96 = ( 108 , 108 ) 9 n 3 51 n 2 + 72 n

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