EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 17, No. 2, 2024, 690-709 ISSN 1307-5543 – ejpam.com Published by New York Business Global Graphical Invariants for some Transformed Networks Nawaf Ali1, Tarek Khalifa1, Hifza Iqbal2, Muhammad Haroon Aftab2,∗, Kamel Jebreen3,4,5, Humira Jamil2, Hassan Kanj1 1 College of Engineering and Technology, /American University of the Middle East, Egaila 54200, Kuwait 2 Department of Mathematics and Statistics, The University of Lahore, Lahore 54500, Pakistan 3 Department of Mathematics, /Palestine Technical University-Kadoorie, Hebron, Palestine 4 Department of Mathematics, /An-Najah National University, Nablus, Palestine 5 Biostatistics and Clinical Research Department, /University Hospital, Lariboisière, AP-HP, Universite’ Paris, France Abstract. A topological index is a numerical character associated with a graph that is invariant under graph isomorphism and describe the graphs topology. There are several graph operations that may be used to change it into a new structure, such as constructing a stellation, bounded dual, complement, subdivided, line graph, minor, dual, and medial. In this paper, we construct trans- formed networks from the concealled non-kekulean benzenoid hydrocarbon structure by applying stellation and bounded dual operations, further we study their degree based topological properties by appropriately labeling the graph. Degree based topological indices are playing significant role among other types of indices in chemical, pharmaceutical and bio-informatics industry, since they corelate the structure with its physicochemical properties. 2020 Mathematics Subject Classifications: 05C08, 05C92, 37F20 Key Words and Phrases: Topological Indices; Concealled Non-Kekulean Benezenoid Hydrocar- bon; Sum Connectivity Index; General Sum Connectivity Index; Atomic Bond Connectivity Index; Geometric Arithmetic Index 1. Introduction The subject of mathematics known as graph theory deals with network of points con- nected by lines. Graph theory began as a fun way to solve math problems, but it has now evolved into a major field of mathematics with applications in chemistry, operations ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v17i2.5078 Email addresses: nawaf.ali@aum.edu.kw (N. Ali), tarek.khalifa@aum.edu.kw (T. Khalifa), iqbalhifza3@gmail.com (H. Iqbal), muhammadharoonaftab@gmail.com (M. H. Aftab), k.jebreen@yahoo.com (K. Jebreen), huumaira2690@gmail.com (H. Jamil), hassan.kanj@aum.edu.kw (H. Kanj). https://www.ejpam.com 690 © 2024 EJPAM All rights reserved. M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 691 research, social sciences, and computer science. An important use of (connected and undi- rected) graphs is the representation of an atomic structure by a graph where the vertices represent atoms and the edges indicate bonding, which is explored in the field of chemical graph theory. These indices are playing vital role to study various networks since they help researchers in QSPR (Quantitative Structure Property Relationship) and QSAR (Quan- titative Structure Activity Relationship) study. Alikhani et al. calculated the atom-bond connectivity index of some families of dendrimers [3]. Babujee et al. worked on topological indices and new graph structures [4]. Farahani worked on a new version of Zagreb index of circumcoronene series of benzenoid and cal- culated some connectivity indices of different classes of graphs [6], [7], [8], [9]. Hayat et al. calculated some degree-based topological indices of certain nanotubes and networks [11], [14]. Few networks are discussed in [16], [13], [12], [42], [29], [15], [23]. Ma et al. studied the energy and operations of graphs [28]. Randic also calculated the benzenoid rings resonance energies and local aromaticity of benzenoid hydrocarbons [31]. Saleem worked on retractions and homomorphisms on some operations of graphs [33]. Siddiqui et al. worked on Zagreb indices and Zagreb polynomials of some nanostar dendrimers [34]. Yu et al. defined indices through M-polynomial [37]. There are thousands of topological indices developed over the decades in the field of chem- ical graph theory. Since different indices deal with different structure properties, others give better estimation. The results of our study are novel, motivated by the application of indices in the advanced material technology, to form new materials and study its various properties. If dθ and dψ are the degrees of the vertices θ and ψ, respectively in K and θψ∈ E(K) then following are the formulae of different degree based topological indices which are computed in this paper. The General Randic Index was defined as [30] Rα(K) = ∑ θψ∈E(K) (dθdψ) α (1) The Sum Connectivity Index was defined as [40] χ(K) = ∑ θψ∈E(K) (dθ + dψ) −1/2 (2) The General Sum Connectivity Index was defined by Zhou [41] χα(K) = ∑ θψ∈E(K) (dθ + dψ) α (3) Ranjini in 2013, stated the Redefined First, Second and Third Zagreb Indices [32] ReZG1(K) = ∑ θψ∈E(K) ( dθ + dψ dθdψ ) (4) ReZG2(K) = ∑ θψ∈E(K) ( dθdψ dθ + dψ ) (5) M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 692 ReZG3(K) = ∑ θψ∈E(K) (dθdψ)(dθ + dψ) (6) Whereas in 2010 the Atomic Bond Connectivity Index was defined as [5] ABC(K) = ∑ θψ∈E(K) √ dθ + dψ − 2 dθdψ (7) Furtula stated Geometric Arithmetic Index as [10] GA(K) = ∑ θψ∈E(K) 2 √ dθdψ dθ + dψ (8) In 2015, the General Version of Harmonic Index was defined [36] Hk(K) = ∑ θψ∈E(K) ( 2 dθ + dψ )k (9) 2. Results and Discussion To understand the concept of stellation and bounded dual, see the following Figures. The Figure 1, is of single benzene ring. Similarly, Figure 2, shows stellation on benzene ring and Figure 3, shows stellation plus bounded dual on the benzene rings of concealled non- kekulean benezenoid hydrocarbon, respectively. 1 Figure 1: Single Ring of Concealled Non-Kekulean Benzenoid Hydrocarbon. Let G1 be the simple and undirected molecular graph when stellation operation is applied on concealled non-kekulean benzenoid hydrocarbon and Figure 4 shows the stellation network for n = 6. Six different types of edges of graph, G1 for n≥4 and their count are given in Table 1. Following are some results of topological indices for G1. M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 693 1 Figure 2: Stellation is Blue. Single Ring of Concealled Non-Kekulean Benzenoid Hydrocarbon. 1 Figure 3: Stellation is Blue. Bounded Dual is Red. Both Operations are on Couple of Benzene Ring. Table 1: Types and number of edges Types of Edges Number of Edges (3 , 3) 8 (3 , 5) 12+4n (3 , 6) 14+2n (5 , 6) 24+6n (6 , 6) 35n-40 (5 , 5) 2 1 Figure 4: Stellation Operation on Concealled Non-Kekulean Benzenoid Hydrocarbon for n = 6. M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 694 Theorem 2.1 For G1, the General Randic Index, Sum Connectivity Index, General Sum Connectivity Index are as follows, respectively. i) Rα(G1) = 2 [ (4)32α + (2)31+α5α + (7)2α32α + 22+α31+α5α − (5)22(1+α)32α +52α ] + n [ (4)3α5α + 21+α32α + 21+α31+α5α + (35)22α32α ] ii) χ(G1) = 4 √ 6 + 9 √ 2 + 14− 20 √ 3 3 + 24√ 11 + √ 2 5 + n (√ 2 + 2 3 + 6√ 11 + 35 2 √ 3 ) iii) χα(G1) = 2 [ 22+α3α + (3)21+3α + (7)32α + (12)11α − (5)22(1+α)3α + 2α5α ] +n [ 22+3α + (2)32α + (6)11α + (35)22α3α ] Proof i) According to Equation (1) Rα(G1) = ∑ θψ∈E(G1) (dθdψ) α By using the information given in Table 1. Rα(G1) = (8)(3× 3)α + (12 + 4n)(3× 5)α + (14 + 2n)(3× 6)α + (24 + 6n)(5× 6)α +(35n− 40)(6× 6)α + (2)(5× 5)α = 2 [ (4)32α + (2)31+α5α + (7)2α32α + 22+α31+α5α − (5)22(1+α)32α + 52α ] +n [ (4)3α5α + 21+α32α + 21+α31+α5α + (35)22α32α ] ii) According to Equation (2) χ(G1) = ∑ θψ∈E(G1) (dθ + dψ) −1/2 By using the information given in Table 1. χ(G1) = (8)(3 + 3)−1/2 + (12 + 4n)(3 + 5)−1/2 + (14 + 2n)(3 + 6)−1/2 + (24 + 6n)(5 +6)−1/2 + (35n− 40)(6 + 6)−1/2 + (2)(5 + 5)−1/2 = 4 √ 6 + 9 √ 2 + 14− 20 √ 3 3 + 24√ 11 + √ 2 5 + n (√ 2 + 2 3 + 6√ 11 + 35 2 √ 3 ) M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 695 iii) According to Equation (3) χα(G1) = ∑ θψ∈E(G1) (dθ + dψ) α By using the information given in Table 1. χα(G1) = (8)(3 + 3)α + (12 + 4n)(3 + 5)α + (14 + 2n)(3 + 6)α + (24 + 6n)(5 + 6)α +(35n− 40)(6 + 6)α + (2)(5 + 5)α = 2 [ 22+α3α + (3)21+3α + (7)32α + (12)11α − (5)22(1+α)3α + 2α5α ] + n [ 22+3α +(2)32α + (6)11α + (35)22α3α ] Theorem 2.2 For G1, the 1st Zagreb Index, 2nd Zagreb Index and 3rd Zagreb Index are as follows, respectively. i) ReZG1(G1) = 15 + 17n ii) ReZG2(G1) = 285 + 2923n 22 iii) ReZG3(G1) = 17904n− 4720 Proof i) According to Equation (4) ReZG1(G1) = ∑ θψ∈E(G1) ( dθ + dψ dθdψ ) By using the information given in Table 1. ReZG1(G1) = (8) ( 3 + 3 3× 3 ) + (12 + 4n) ( 3 + 5 3× 5 ) + (14 + 2n) ( 3 + 6 3× 6 ) +(24 + 6n) ( 5 + 6 5× 6 ) + (35n− 40) ( 6 + 6 6× 6 ) + (2) ( 5 + 5 5× 5 ) = 15 + 17n ii) According to Equation (5) ReZG2(G1) = ∑ θψ∈E(G1) ( dθdψ dθ + dψ ) M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 696 By using the information given in Table 1. ReZG2(G1) = (8) ( 3× 3 3 + 3 ) + (12 + 4n) ( 3× 5 3 + 5 ) + (14 + 2n) ( 3× 6 3 + 6 ) + (24 + 6n) ( 5× 6 5 + 6 ) +(35n− 40) ( 6× 6 6 + 6 ) + (2) ( 5× 5 5 + 5 ) = 285 + 2923n 22 iii) According to Equation (6) ReZG3(G1) = ∑ θψ∈E(G1) (dθdψ)(dθ + dψ) By using the information given in Table 1. ReZG3(G1) = (8)(3× 3)(3 + 3) + (12 + 4n)(3× 5)(3 + 5) + (14 + 2n)(3× 6)(3 + 6) + (24 +6n)(5× 6)(5 + 6) + (35n− 40)(6× 6)(6 + 6) + (2)(5× 5)(5 + 5) = 17904n− 4720 Theorem 2.3 For G1, the Atomic Bond Connectivity, Geometric Arithmetic Index and General Ver- sion of Harmonic Index are as follows, respectively. i) ABC(G1) = 1√ 2 [ 2 ( 40 √ 2 + 35 √ 7− 64 √ 5 + 36 √ 15 + 12 15 ) +n ( 199 + 18 √ 3 + 2 √ 35 3 √ 5 )] ii) GA(G1) = 99 (√ 15− 10 ) + 4 √ 2 ( 77 + 36 √ 15 ) 33 +n [ 11(105 + 4 √ 2) + 3 √ 15(11 + 12 √ 2) 33 ] iii) Hk(G1) = 23 3k ( 2k − 5 2k ) + (3)22(1−k) + (7)21+k 32k + (3)23+k 11k + 2 5k +n ( 22(1−k) + 21+k 32k + (3)21+k 11k + 35 2k3k ) M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 697 Proof i) According to Equation (7) ABC(G1) = ∑ θψ∈E(G1) √ dθ + dψ − 2 dθdψ By using the information given in Table 1. ABC(G1) = (8) √ 3 + 3− 2 3× 3 + (12 + 4n) √ 3 + 5− 2 3× 5 + (14 + 2n) √ 3 + 6− 2 3× 6 +(24 + 6n) √ 5 + 6− 2 5× 6 + (35n− 40) √ 6 + 6− 2 6× 6 + (2) √ 5 + 5− 2 5× 5 = 1√ 2 [ 2 ( 40 √ 2 + 35 √ 7− 64 √ 5 + 36 √ 15 + 12 15 ) + n ( 199 + 18 √ 3 + 2 √ 35 3 √ 5 )] ii) According to Equation (8) GA(G1) = ∑ θψ∈E(G1) 2 √ dθdψ dθ + dψ By using the information given in Table 1. GA(G1) = (8) 2 √ 3× 3 3 + 3 + (12 + 4n) 2 √ 3× 5 3 + 5 + (14 + 2n) 2 √ 3× 6 3 + 6 + (24 + 6n) 2 √ 5× 6 5 + 6 +(35n− 40) 2 √ 6× 6 6 + 6 + (2) 2 √ 5× 5 5 + 5 = 99 (√ 15− 10 ) + 4 √ 2 ( 77 + 36 √ 15 ) 33 + n [ 11(105 + 4 √ 2) + 3 √ 15(11 + 12 √ 2) 33 ] iii) According to Equation (9) Hk(G1) = ∑ θψ∈E(G1) ( 2 dθ + dψ )k By using the information given in Table 1. Hk(G1) = (8) ( 2 3 + 3 )k + (12 + 4n) ( 2 3 + 5 )k + (14 + 2n) ( 2 3 + 6 )k M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 698 +(24 + 6n) ( 2 5 + 6 )k + (35n− 40) ( 2 6 + 6 )k + (2) ( 2 5 + 5 )k = 23 3k ( 2k − 5 2k ) + (3)22(1−k) + (7)21+k 32k + (3)23+k 11k + 2 5k +n ( 22(1−k) + 21+k 32k + (3)21+k 11k + 35 2k3k ) Now, we are going to calculate topological indices when both the bounded dual and stel- lation operations are applied on the graph of concealled non-kekulean benzenoid hydro- carbon, say G2. Let G2 be the simple and undirected transformed network and following are twenty types of edges for G2, n≥4. These types of edges and their count are given in Table 2 and Figure 5 shows the stellation and bounded dual on single structure for n = 7. Table 2: Types and number of edges Types of Edges Number of Edges (3 , 3) 8 (3 , 5) 4n+12 (5 , 6) 8+2n (8 , 3) 12 (6 , 8) 4 (8 , 5) 8 (10 , 10) 2n (3 , 10) 2+2n (10 , 6) 6+6n (10 , 8) 8 (5 , 10) 4+4n (6 , 6) 11n-16 (5 , 5) 2 (6 , 12) 18n-42 (10 , 12) 4n-4 (5 , 11) 4 (6 , 11) 8 (12 , 11) 6 (10 , 11) 4 (12 , 12) 7n-22 Theorem 2.4 For G2, the General Randic Index, Sum Connectivity Index, General Sum Connectiv- ity Index are as follows, respectively. M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 699 1 Figure 5: Stellation and Bounded Dual on the Same Structure of Concealled Non-Kekulean Benzenoid Hydro- carbon for n = 7. i) Rα(G2) = 32α8 + 31+α5α4 + 23+α3α5α + 22+3α31+α + 22(1+2α)3α +23(1+α)5α + 21+α3α5α + 21+2α31+α5α + 23+4α5α + 22+α52α −22(2+α)32α + 52α2− 21+3α31+2α7− 22+3α3α5α + 5α11α4 +23+α3α11α + 21+2α31+α11α + 22+α5α11α − 21+4α32α11 +n[3α5α4 + 21+α5α3α + 21+2α52α + 21+α3α5α + 21+2α31+α5α +22+α52α + 22α32α11 + 21+3α32(1+α) + 22+3α3α5α + 32α24α7] ii) χ(G2) = 4√ 3 [√ 2 + √ 2 3 + √ 1 5 − 2 + √ 1 7 − 11 4 √ 2 ] + 1√ 11 [ 20 + √ 2 ] + 10√ 13 + 2 √ 2 7 + √ 2 5 + 8√ 17 − 4 √ 2 + 5 2 + n [ 2√ 11 ( 1 + √ 2 ) + 1√ 5 (√ 3 + 4√ 3 ) + 1 2 √ 3 ( 11 √ 2 + 7√ 2 ) + 4 √ 2 + 2√ 13 + 3 2 ] iii) χα(G2) = 23+α3α + 22+3α3 + 11α8 + 11α12 + 22+α7α + 13α8 + 13α2 +21+4α3 + 23+α32α + 5α3α4− 22(2+α)3α + 21+α5α − 31+2α21+α7 −22+α11α + 22(1+2α) + 17α8 + 23α6 + 7α3α4− 21+3α3α11 +n [ 22+3α + 11α2 + 21+2α5α + 13α2 + 21+4α3 + 5α3α4 +22α3α11 + 32(1+α)21+α + 22+α11α + 23α3α7 ] . M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 700 Proof i) According to Equation (1) Rα(G2) = ∑ θψ∈E(G2) (dθdψ) α By using types of edges given in Table 2, we get. Rα(G2) = (8)(3× 3)α + (12 + 4n)(3× 5)α + (8 + 2n)(5× 6)α + (12)(8× 3)α + (4)(6× 8)α +(8)(8× 5)α + (2n)(10× 10)α + (2 + 2n)(3× 10)α + (6 + 6n)(10× 6)α +(8)(10× 8)α + (4 + 4n)(5× 10)α + (11n− 16)(6× 6)α + (2)(5× 5)α +(18n− 42)(6× 12)α + (4n− 4)(10× 12)α + (4)(5× 11)α + (8)(6× 11)α +(6)(12× 11)α + (4)(10× 11)α + (7n− 22)(12× 12)α = 32α8 + 31+α5α4 + 23+α3α5α + 22+3α31+α + 22(1+2α)3α + 23(1+α)5α + 21+α3α5α +21+2α31+α5α + 23+4α5α + 22+α52α − 22(2+α)32α + 52α2− 21+3α31+2α7 −22+3α3α5α + 5α11α4 + 23+α3α11α + 21+2α31+α11α + 22+α5α11α −21+4α32α11 + n[3α5α4 + 21+α5α3α + 21+2α52α + 21+α3α5α + 21+2α31+α5α +22+α52α + 22α32α11 + 21+3α32(1+α) + 22+3α3α5α + 32α24α7]. ii) According to Equation (2) χ(G2) = ∑ θψ∈E(G2) (dθ + dψ) −1/2 By using types of edges given in Table 2, we get. χ(G2) = (8)(3 + 3)−1/2 + (4n+ 12)(3 + 5)−1/2 + (8 + 2n)(5 + 6)−1/2 + (12)(8 + 3)−1/2 +(4)(6 + 8)−1/2 + (8)(8 + 5)−1/2 + (2n)(10 + 10)−1/2 + (2 + 2n)(3 + 10)−1/2 +(6 + 6n)(10 + 6)−1/2 + (8)(10 + 8)−1/2 + (4 + 4n)(5 + 10)−1/2 + (11n− 16) ×(6 + 6)−1/2 + (2)(5 + 5)−1/2 + (18n− 42)(6 + 12)−1/2 + (4n− 4)(10 + 12)−1/2 +(4)(5 + 11)−1/2 + (8)(6 + 11)−1/2 + (6)(12 + 11)−1/2 + (4)(10 + 11)−1/2 +(7n− 22)(12 + 12)−1/2 = 4√ 3 [√ 2 + √ 2 3 + √ 1 5 − 2 + √ 1 7 − 11 4 √ 2 ] + 1√ 11 [ 20 + √ 2 ] + 10√ 13 + 2 √ 2 7 + √ 2 5 + 8√ 17 − 4 √ 2 + 5 2 + n [ 2√ 11 ( 1 + √ 2 ) + 1√ 5 ( 1 + 4√ 3 ) + 1 2 √ 3 ( 11 + 7√ 2 ) + 4 √ 2 + 2√ 13 + 3 2 ] . M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 701 iii) According to Equation (3) χα(G2) = ∑ θψ∈E(G2) (dθ + dψ) α By using types of edges given in Table 2, we get. χα(G2) = (8)(3 + 3)α + (4n+ 12)(3 + 5)α + (8 + 2n)(5 + 6)α + (12)(8 + 3)α + (4)(6 +8)α + (8)(8 + 5)α + (2n)(10 + 10)α + (2 + 2n)(3 + 10)α + (6 + 6n)(10 + 6)α +(8)(10 + 8)α + (4 + 4n)(5 + 10)α + (11n− 16)(6 + 6)α + (2)(5 + 5)α + (18n −42)(6 + 12)α + (4n− 4)(10 + 12)α + (4)(5 + 11)α + (8)(6 + 11)α + (6)(12 +11)α + (4)(10 + 11)α + (7n− 22)(12 + 12)α = 23+α3α + 22+3α3 + 11α8 + 11α12 + 22+α7α + 13α8 + 13α2 + 21+4α3 + 23+α32α +5α3α4− 22(2+α)3α + 21+α5α − 31+2α21+α7− 22+α11α + 22(1+2α) + 17α8 +23α6 + 7α3α4− 21+3α3α11 + n [ 22+3α + 11α2 + 21+2α5α + 13α2 + 21+4α3 +5α3α4 + 22α3α11 + 32(1+α)21+α + 22+α11α + 23α3α7 ] . Theorem 2.5 For G2, the 1st Zagreb Index, 2nd Zagreb Index and 3rd Zagreb Index are as follows, respectively. i) ReZG1(G2) = 15 + 17n ii) ReZG2(G2) = 232.22n− 67.79 iii) ReZG3(G2) = 77512n− 71896. Proof i) According to Equation (4) ReZG1(G2) = ∑ θψ∈E(G2) ( dθ + dψ dθdψ ) By using types of edges given in Table 2, we get. ReZG1(G2) = (8) ( 3 + 3 3× 3 ) + (4n+ 12) ( 3 + 5 3× 5 ) + (8 + 2n) ( 5 + 6 5× 6 ) + (12) ( 8 + 3 8× 3 ) +(4) ( 6 + 8 6× 8 ) + (8) ( 8 + 5 8× 5 ) + (2n) ( 10 + 10 10× 10 ) + (2 + 2n) ( 3 + 10 3× 10 ) M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 702 +(6 + 6n) ( 10 + 6 10× 6 ) + (8) ( 10 + 8 10× 8 ) + (4 + 4n) ( 5 + 10 5× 10 ) + (11n− 16) ( 6 + 6 6× 6 ) +(2) ( 5 + 5 5× 5 ) + (18n− 42) ( 6 + 12 6× 12 ) + (4n− 4) ( 10 + 12 10× 12 ) + (4) ( 5 + 11 5× 11 ) +(8) ( 6 + 11 6× 11 ) + (6) ( 12 + 11 12× 11 ) + (4) ( 10 + 11 10× 11 ) + (7n− 22) ( 12 + 12 12× 12 ) = 14850 990 + n ( 510 30 ) = 15 + 17n. ii) According to Equation (5) ReZG2(G2) = ∑ θψ∈E(G2) ( dθdψ dθ + dψ ) By using types of edges given in Table 2, we get. ReZG2(G2) = (8) ( 3× 3 3 + 3 ) + (4n+ 12) ( 3× 5 3 + 5 ) + (8 + 2n) ( 5× 6 5 + 6 ) + (12) ( 8× 3 8 + 3 ) +(4) ( 6× 8 6 + 8 ) + (8) ( 8× 5 8 + 5 ) + (2n) ( 10× 10 10 + 10 ) + (2 + 2n) ( 3× 10 3 + 10 ) + (6 +6n) ( 10× 6 10 + 6 ) + (8) ( 10× 8 10 + 8 ) + (4 + 4n) ( 5× 10 5 + 10 ) + (11n− 16) ( 6× 6 6 + 6 ) +(2) ( 5× 5 5 + 5 ) + (18n− 42) ( 6× 12 6 + 12 ) + (4n− 4) ( 10× 12 10 + 12 ) + (4) ( 5× 11 5 + 11 ) +(8) ( 6× 11 6 + 11 ) + (6) ( 12× 11 12 + 11 ) + (4) ( 10× 11 10 + 11 ) + (7n− 22) ( 12× 12 12 + 12 ) = −67.79 + 232.22n = 232.22n− 67.79. iii) According to Equation (6) ReZG3(G2) = ∑ θψ∈E(G2) (dθdψ)(dθ + dψ) By using types of edges given in Table 2, we get. ReZG3(G2) = (8)(3× 3)(3 + 3) + (4n+ 12)(3× 5)(3 + 5) + (8 + 2n)(5× 6)(5 + 6) +(12)(8× 3)(8 + 3) + (4)(6× 8)(6 + 8) + (8)(8× 5)(8 + 5) + (2n)(10 M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 703 ×10)(10 + 10) + (2 + 2n)(3× 10)(3 + 10) + (6 + 6n)(10× 6)(10 + 6) +(8)(8× 10)(8 + 10) + (4 + 4n)(5× 10)(5 + 10) + (11n− 16)(6× 6)(6 +6) + (2)(5× 5)(5 + 5) + (18n− 42)(6× 12)(6 + 12) + (4n− 4)(10 ×12)(10 + 12) + (4)(5× 11)(5 + 11) + (8)(6× 11)(6 + 11) + (6)(12 ×11)(12 + 11) + (4)(10× 11)(10 + 11) + (7n− 22)(12× 12)(12 + 12) = −71896 + 77512n = 77512n− 71896. Theorem 2.6 For G2, the Atomic Bond Connectivity, Geometric Arithmetic Index and General Ver- sion of Harmonic Index are as follows, respectively. i) ABC(G2) = 2(11− 4 √ 10) 3 + 8 √ 3(3 + √ 2) + 2 √ 11(2 √ 3 + 1) + 2(12 + 7 √ 5 + 3 √ 7)√ 30 + 2(444 + 12 √ 13− 55 √ 11) 30 √ 2 + √ 7(8 + 3 √ 10) + 4( √ 19 + 10)√ 110 +n [ 2(4 √ 3 + 3 + √ 11 + 3 √ 7)√ 30 + 2(33 + 2 √ 13) 5 √ 2 + √ 2(22 √ 5 + 7 √ 11) 12 + 4√ 6 ] . ii) GA(G2) = 3 √ 15− 28− 28 √ 2 + 8 √ 6(2 √ 5 + 6− √ 5) 11 + 8(6 √ 3 + √ 110) 21 + 4 √ 10(8 + √ 3) 13 + √ 5(3 √ 3 + √ 11) 2 + 8(4 √ 5 + 3 √ 2) 9 + 16 √ 66 17 + 24 √ 33 23 + n [√ 15 + 20 + 12 √ 2 + 12 √ 30 11 + 4 √ 30 13 + 9 √ 15 + 16 √ 2 6 ] . iii) Hk(G2) = 22+k5 11k + 21+k5 13k + 22(1+k) − 5k16 + 21+k3k 2k3k5k + 21+2k − 7k11 2k−13k7k + 21+k3 + 5 23k−1 + 2(3k4− 17) 32k + 4 7k − 4 11k + 23+k 17k + 21+k3 23k +n [ 3k2 + 22(1+k) + 5k11 2k3k5k + 21+k + 3 23k−1 + 4(2k−1 + 1) 11k + 22k+13 + 3k−17 32k−122k + 21+k 13k ] . M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 704 Proof i) According to Equation (7) ABC(G2) = ∑ θψ∈E(G2) √ dθ + dψ − 2 dθdψ By using types of edges given in Table 2, we get. ABC(G2) = (8) √ 3 + 3− 2 3× 3 + (4n+ 12) √ 3 + 5− 2 3× 5 + (8 + 2n) √ 5 + 6− 2 5× 6 + (12) √ 8 + 3− 2 8× 3 +(4) √ 6 + 8− 2 6× 8 + (8) √ 8 + 5− 2 8× 5 + (2n) √ 10 + 10− 2 10× 10 + (2 + 2n) √ 3 + 10− 2 3× 10 +(6 + 6n) √ 10 + 6− 2 10× 6 + (8) √ 10 + 8− 2 10× 8 + (4 + 4n) √ 5 + 10− 2 5× 10 +(11n− 16) √ 6 + 6− 2 6× 6 + (2) √ 5 + 5− 2 5× 5 + (18n− 42) √ 6 + 12− 2 6× 12 +(4n− 4)) √ 10 + 12− 2 10× 12 + (4) √ 5 + 11− 2 5× 11 + (8) √ 6 + 11− 2 6× 11 + (6) √ 12 + 11− 2 12× 11 +(4) √ 10 + 11− 2 10× 11 + (7n− 22) √ 12 + 12− 2 12× 12 = 2(11− 4 √ 10) 3 + 8 √ 3(3 + √ 2) + 2 √ 11(2 √ 3 + 1) + 2(12 + 7 √ 5 + 3 √ 7)√ 30 + 2(444 + 12 √ 13− 55 √ 11) 30 √ 2 + √ 7(8 + 3 √ 10) + 4( √ 19 + 10)√ 110 +n [ 2(4 √ 3 + 3 + √ 11 + 3 √ 7)√ 30 + 2(33 + 2 √ 13) 5 √ 2 + √ 2(22 √ 5 + 7 √ 11) 12 + 4√ 6 ] . ii) According to Equation (8) GA(G2) = ∑ θψ∈E(G2) 2 √ dθdψ dθ + dψ By using types of edges given in Table 2, we get. GA(G2) = (8) ( 2 √ 3× 3 3 + 3 ) + (4n+ 12) ( 2 √ 3× 5 3 + 5 ) + (8 + 2n) ( 2 √ 5× 6 5 + 6 ) + (12) ( 2 √ 8× 3 8 + 3 ) +(4) ( 2 √ 8× 6 8 + 6 ) + (8) ( 2 √ 8× 5 8 + 5 ) + (2n) ( 2 √ 10× 10 10 + 10 ) + (2 + 2n) ( 2 √ 3× 10 3 + 10 ) +(6 + 6n) ( 2 √ 10× 6 10 + 6 ) + (8) ( 2 √ 10× 8 10 + 8 ) + (4 + 4n) ( 2 √ 5× 10 5 + 10 ) M. H. Aftab et al. / Eur. J. Pure Appl. Math, 17 (2) (2024), 690-709 705 +(11n− 16) ( 2 √ 6× 6 6 + 6 ) + (2) ( 2 √ 5× 5 5 + 5 ) + (18n− 42) ( 2 √ 6× 12 6 + 12 ) +(4n− 4) ( 2 √ 10× 12 10 + 12 ) + (4) ( 2 √ 5× 11 5 + 11 ) + (8) ( 2 √ 6× 11 6 + 11 ) + (6) ( 2 √ 12× 11 12 + 11 ) +(4) ( 2 √ 10× 11 10 + 11 ) + (7n− 22) ( 2 √ 12× 12 12 + 12 ) = 3 √ 15− 28− 28 √ 2 + 8 √ 6(2 √ 5 + 6− √ 5) 11 + 8(6 √ 3 + √ 110) 21 + 4 √ 10(8 + √ 3) 13 + √ 5(3 √ 3 + √ 11) 2 + 8(4 √ 5 + 3 √ 2) 9 + 16 √ 66 17 + 24 √ 33 23 + n [√ 15 + 20 + 12 √ 2 + 12 √ 30 11 + 4 √ 30 13 + 9 √ 15 + 16 √ 2 6 ] . iii) According to Equation (9) Hk(G2) = ∑ θψ∈E(G2) ( 2 dθ + dψ )k By using types of edges given in Table 2, we get. Hk(G2) = (8) ( 2 3 + 3 )k + (4n+ 12) ( 2 3 + 5 )k + (8 + 2n) ( 2 5 + 6 )k + (12) ( 2 8 + 3 )k +(4) ( 2 6 + 8 )k + (8) ( 2 8 + 5 )k + (2n) ( 2 10 + 10 )k + (2 + 2n) ( 2 3 + 10 )k +(6 + 6n) ( 2 10 + 6 )k + (8) ( 2 10 + 8 )k + (4 + 4n) ( 2 5 + 10 )k + (11n− 16) ( 2 6 + 6 )k +(2) ( 2 5 + 5 )k + (18n− 42) ( 2 6 + 12 )k + (4n− 4) ( 2 10 + 12 )k + (4) ( 2 5 + 11 )k +(8) ( 2 6 + 11 )k + (6) ( 2 12 + 11 )k + (4) ( 2 10 + 11 )k + (7n− 22) ( 2 12 + 12 )k = 22+k5 11k + 21+k5 13k + 22(1+k) − 5k16 + 21+k3k 2k3k5k + 21+2k − 7k11 2k−13k7k + 21+k3 + 5 23k−1 + 2(3k4− 17) 32k + 4 7k − 4 11k + 23+k 17k + 21+k3 23k + n [ 3k2 + 22(1+k) + 5k11 2k3k5k + 21+k + 3 23k−1 + 4(2k−1 + 1) 11k + 22k+13 + 3k−17 32k−122k + 21+k 13k ] . REFERENCES 706 3. Conclusions In this paper, certain degree based topological indices [1, 2, 18–21, 25–27], namely; general Randic index, sum connectivity index, general sum connectivity index, first, sec- ond and third Zagreb indices, atomic bond connectivity index, geometric arithmetic index, general version of harmonic index are computed for transformed structures by applying stellation and bounded dual operations, derived networks are named G1 and G2. Analyti- cally closed formulae of above mentioned topological indices for these networks are stated. The results provide a basis to understand a deep topology for G1 and G2 graphs. The compelling factor of these topological indices is that they are structure invariant. In future, the interested reader is encouraged to design some new architectures/networks and study their topological properties through these numerical descriptors [17, 22, 24, 35, 38, 39]. 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