Canadian Journal of Chemistry. On molecular topological properties of diamond like networks

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1 anaian Journal of hemistry On molecular topological properties of iamon like networks Journal: anaian Journal of hemistry Manuscript I cjc r Manuscript Type: Article ate Submitte by the Author: 28-Apr-207 omplete List of Authors: Imran, Muhamma; Unite Arab mirates University, epartment of Mathematical Sciences; National University of Sciences an Technology, School of Natural Sciences aig, A. Q.; OMSATS Insitute of Information Technology, Attock, Pakistan, Mathematics Siiqui, Hafiz Muhamma Afzal; OMSATS Institute of Information Technology - MA Jinnah ampus, epartment of Mathematics Sarawar, Rabia; Government ollege University aisalaba Is the invite manuscript for consieration in a Special Issue?: Keywor: N/A Generalizes Aztec iamon, xtene Aztec iamon, Thir connectivity inex, Thir sum-connectivity-inex

2 Page of 23 anaian Journal of hemistry On molecular topological properties of iamon like networks a,b Muhamma Imran, c Abul Quair aig, Hafiz Muhamma Afzal Siiqui, e Rabia Sarwar, a epartment of Mathematical Sciences, Unite Arab mirates University, P. O. ox 555, Al Ain, Unite Arab mirates b School of Natural Sciences, National University of Sciences an Technology, H-2, Islamaba, Pakistan c epartment of Mathematics, OMSATS Institute of Information Technology, Attock, Pakistan epartment of Mathematics, OMSATS Institute of Information Technology Lahore, Pakistan e epartment of Mathematics, Government ollege University aisalaba (GU, Pakistan mail: {imranhab, aqbaig, hmasiiqui, rabiasarwar80}@gmail.com This research is supporte by the supporte by the Start Up Research Grant 206 of Unite Arab mirates University, Al Ain, Unite Arab mirates via Grant No. G an by the grant of Higher ucation ommission of Pakistan Ref. No /SRGP/R&/H/206.

3 anaian Journal of hemistry Page 2 of 23 2 M. Imran, A. Q. aig, H. M. A. Siiqui, R. Sarwar Abstract. The Ranić (prouct connectivity inex an its erivative calle the sum-connectivity inex are well known topological inices an these both escriptors correlate well among themselves an with the π-electronic energies of benzenoi hyrocarbons. The general n-connectivity of a molecular graph G is efine as n χ(g= an the n-sum con- i i2... in nectivity of a molecular graph G is efine as n X(G = v i v i2 v i3...v in i i2... in, v i v i2 v i3...v in where the paths of length n in G are enote by v i, v i2...v in an the egree of each vertex v i is enote by i. In this paper, we iscuss thir connectivity an thir sum-connectivity inices of iamon like networks an compute analytical close results of these inices for iamon like networks. Keywors: Generalizes Aztec iamon, xtene Aztec iamon; Thir connectivity inex; Thir sum-connectivity-inex.

4 Page 3 of 23 anaian Journal of hemistry Introuction an preliminary results Molecular topological properties of iamon like networks 3 Graph theory is proficient for moeling an esigning of chemical structures an complex networks an to preict their properties. The chemical graph theory applies the tools from graph theory to mathematical moeling of molecular phenomena, which is helpful for the stuy of molecular structure an molecular moeling. This theory play a vital role in the fiel of theoretical chemical sciences. hemical compouns have a variety of applications in chemical graph theory, rug esign, an in nanotechnology etc. The manipulation an examination of chemical structural information is mae conceivable by using molecular escriptors. A great variety of topological inices are stuie an use in theoretical chemistry, pharmaceutical researchers. A chemical structure can be represente by using graph theory, where vertices enote atoms an eges enote molecular bon. A topological inex is a numeric number which inicates some useful information about molecular structure. It is the numerical invariants of a molecular graph an are useful to correlate with their bioactivity an physico-chemical properties. Researchers have foun topological inex to be powerful an useful tool in the escription of molecular structure. A graph G(V, is a pair of V (G an (G. Where V (G represent the set of vertices an (G represent the set of eges. The orer of a graph is n an it is enote as V (G = n. The size of a graph is m an it is enote as (G = m. The general n-connectivity inex of a graph G is enote as n χ(g = i i2... in v i v i2 v i3...v in. ( Where the paths of length n in G are enote by v i, v i2...v in an the egree of vertex v i is enote by i. The first connectivity inex is escribe as χ(g = uv (G u v. (2 In 97, Milan Ranić introuce 3 first connectivity inex. Now it is known as Ranić inex. Many variations of Ranić connectivity inex have been iscusse in,5,8,2,23,2,28,29,3,32. The secon connectivity inex is expresse as Similarly, thir connectivity is expresse as 3 χ(g = 2 χ(g = v i v i2 v i3 v i v i2 v i3 v i i i2 i3. (3 i i2 i3 i. (

5 anaian Journal of hemistry Page of 23 M. Imran, A. Q. aig, H. M. A. Siiqui, R. Sarwar In 2008,. Zhou an N. Trinajstić see,2,3,29,35. evelope the sum-connectivity inex, an it is efine as X(G = uv (G u v. (5 Where the egree of vertices u an v are represente as u an v respectively. Similarly, secon-sumconnectivity is escribe as 2 X(G = i i2 i3. (6 v i v i2 v i3 Where the paths of length 2 in graph G are enote by {v i, v i2, v i3 } an egree of each vertex v i is enote by vi where i 3. Similarly, thir sum-connectivity is expresse as 3 X(G =. (7 i i2 i3 i v i v i2 v i3 v i In general, the n sum-connectivity inex of graph G is given by n X(G = i i2... in v i v i2 v i3...v in. (8 The further stuy of the m-connectivity inex is explaine in 6,7,8,3,5,6,23,26,27,33. The further stuy of the m sum-connectivity inex is clarifie 9,0,,5,6,25,32,3. The topological escriptors of certain networks an nanotubes are stuie in 3,9,20. The iamon like networks have been examine in ifferent ways. The relation of omino tilings an Aztec iamon theorem has been argue in,22. In this paper, we examine thir connectivity an thir sum-connectivity of iamon like networks. 2 Main results for generalize Aztec iamon An Aztec iamon with orer n containing all squares lattice whose centre (x, y satisfy x y n. Here n is fixe, an square lattice contains unit squares with the origin as vertex of egree, so that both x an y are half integer. onsier a path L i with i vertices, where i =, 2, 3,,... The tensor prouct of AZ(2 AZ(3 AZ( AZ(5 ig. : Aztec iamon of ifferent imensions.

6 Page 5 of 23 anaian Journal of hemistry Molecular topological properties of iamon like networks 5 two paths L n an L m enote by L n L m is the graph on n m vertices { ( x, y : x n, y m}. Any two vertices are ajacent if an only if x x y y =. learly, this graph is a isconnecte ig. 2: Generalize Aztec iamon L L. graph having two connecte components. One component is enote as O ( L n L m has the vertices { (x,y : x y is o }. The other component is enote as ( Ln L m has the vertices { ( x,y : x y is even}. irst of all, we iscuss O ( L 2n L 2m an it is calle o generalize Aztec iamon. A O generalize Aztec iamon ig. 3: Types of three eges path connectivity of o generalize Aztec iamon L L. A A A A A ig. : Types of three eges paths of o generalize Aztec iamon L L that start from set A.

7 anaian Journal of hemistry Page 6 of 23 6 M. Imran, A. Q. aig, H. M. A. Siiqui, R. Sarwar Three eges path connectivity is obtaine by twice the set A or 3 χ(o ( L n L m = 2A. Table : ge partition of ege set A of 3 χ(o ( L n L m, where m = n an m, n 6. ijkl arinality of ijkl 2n 2 2 2n ig. 5: Types of three eges paths of o generalize Aztec iamon L L that start from set. Similarly, 3 χ(o ( L n L m = 2. Table 2: ge partition of ege set of 3 χ(o ( L n L m, where m = n an m, n 6. ijkl arinality of ijkl 8 6 6n ig. 6: Types of three eges paths of o generalize Aztec iamon L L that start from set.

8 Page 7 of 23 anaian Journal of hemistry Molecular topological properties of iamon like networks 7 3 χ(o ( L n L m = 2. Table 3: ge partition of ege set of 3 χ(o ( L n L m, where m = n an m, n 6. ijkl arinality of ijkl 2n 6n ig. 7: Types of three eges paths of o generalize Aztec iamon L L that start from set. 3 χ(o ( L n L m = 2. Table : ge partition of ege set of 3 χ(o ( L n L m, where m = n an m, n 6. ijkl arinality of ijkl n ig. 8: Types of three eges paths of o generalize Aztec iamon L L that start from set.

9 anaian Journal of hemistry Page 8 of 23 8 M. Imran, A. Q. aig, H. M. A. Siiqui, R. Sarwar 3 χ(o ( L n L m = (m 3, where m = n an m, n 6. Table 5: ge partition of ege set of 3 χ(o ( L n L m, where m = n an m, n 6. ijkl 22 2 arinality of ijkl n 2 n 2 2mn 2m 6n ig. 9: Types of three eges paths of o generalize Aztec iamon L L that start from set. 3 χ(o ( L n L m = (m where m = n an m, n 6. Table 6: ge partition of ege set of 3 χ(o ( L n L m, where m = n an m, n 6. ijkl arinality of G (i, G (j, G (k, G (l 22 n 6 2 n 6 2 2n 8 2mn 6m 6 The following theorems present analytically close formulas of thir connectivity an thir sumconnectivity inices for 3 χ(o ( L 2n L 2m. Theorem Let 3 χ(o ( L 2n L 2m be the graph, then thir connectivity of the graph is given by 2 [ mn ( 2 m ( 3 2n ( ]. 2 Proof. Using the above Tables [ -6 ] an quation( we have 2 χ(g =. i i2 i3 i v i v i2 v i3 v i Since each path repeate twice therefore, we take

10 Page 9 of 23 anaian Journal of hemistry 2 [ a A a i2 i3 i b i2 i3 i e 2 [( 222 Molecular topological properties of iamon like networks 9 b i2 i3 i c e i2 i3 i f ( ( c i2 i3 i f i2 i3 i ] = ( ( ( ( ] = 2 [( n n 6 ( n n 28 ( 2n 8 2 2mn 6m6 ]. After simplification, we get = 2 [ 2 3 n mn ( 2 m ( 3 2n ( ]. 2 2n 2 ( n ( n 2 2 2mn 2m 6n2 n 6 ( 2 2 8n 32 2 Theorem 2 Let 3 X(O ( L 2n L 2m be the graph, then thir sum connectivity of the graph is given by 2 2 [6mn (22m ( 2 3 9n ( ]. Proof. Using the above Tables [ -6 ] an quation(7 we have 2 χ(g =. i i2 i3 i v i v i2 v i3 v i Since each path repeate twice therefore, we take 2 [ a a A i2 i3 i b i2 i3 i e 2 [( 222 b i2 i3 i c e i2 i3 i f ( ( 2 2 ] = 2 [( ( n n 6 ( n 28 ( 2n 8 2 2mn 6m6 ]. After simplification, we get n ( c i2 i3 i f i2 i3 i ] = ( ( n 2 ( n 2 6n 6 22 ( n mn 2m 6n2 n 6 ( 22 = 2 2 [6mn (22m ( 2 3 9n ( ]. 8n 32 2

11 anaian Journal of hemistry Page 0 of 23 0 M. Imran, A. Q. aig, H. M. A. Siiqui, R. Sarwar 3 Thir connectivity an thir sum-connectivity inices of even generalize Aztec iamon A ven generalize Aztec iamon ig. 0: Types of three eges path of even generalize Aztec iamon L L. In thir connectivity the paths of length in G are enote by v i, v i2, v i3, v i an the egree of each vertex v i is enote by i. Where i =, 2, 3,. 3 χ( ( L n L m = 2A, where m = n an m, n 6. A A A A A ig. : Types of three eges paths of even generalize Aztec iamon L L that start from set A. Table 7: ge partition of ege set A of 3 χ( ( L n L m, where m = n an m, n 6. ijkl arinality of ijkl 8 2 2n 2 2n 2 3 χ( ( L n L m = 2, where m = n an m, n 6. 3 χ( ( L n L m = 2, where m = n an m, n 6. 3 χ( ( L n L m = 2, where m = n an m, n 6.

12 Page of 23 anaian Journal of hemistry Molecular topological properties of iamon like networks 22 ig. 2: Types of three eges paths of even generalize Aztec iamon, that start from set. Table 8: ge partition of ege set of 3 χ( ( L n L m, where m = n an m, n 6. ijkl 22 arinality of ijkl 8 2n 22 2 ig. 3: Types of three eges paths of even generalize Aztec iamon L L that start from set. Table 9: ge partition of ege set of 3 χ( ( L n L m, where m = n an m, n 6. ijkl 22 2 arinality of ijkl 2 20n ig. : Types of three eges paths of even generalize Aztec iamon L L that start from set. Table 0: ge partition of ege set of 3 χ( ( L n L m, where m = n an m, n 6. ijkl arinality of ijkl 2n 2 20n 8 3 χ( ( L n L m = (m 3, where m = n an m, n 6. Table : ge partition of ege set of 3 χ( ( L n L m, where m = n an m, n 6. ijkl arinality of ijkl n 2 8n 2 2n 6 2mn 2n 32

13 anaian Journal of hemistry Page 2 of 23 2 M. Imran, A. Q. aig, H. M. A. Siiqui, R. Sarwar ig. 5: Types of three eges paths of even generalize Aztec iamon that start from set. 3 χ( ( L n L m = (m, where m = n an m, n ig. 6: Types of three eges paths of even generalize Aztec iamon that start from set. Table 2: ge partition of ege set of 3 χ( ( L n L m, where m = n an m, n 6. ijkl arinality of G (i, G (j, G (k, G (l 22 2n 2 2n 2mn 28m 2n 6 ollowing theorems present analytically close formula of thir connectivity an thir sum-connectivity even generalize Aztec iamon. Theorem 3 Let 3 χ( ( L 2n L 2m be the graph, then thir connectivity of the graph is given by 2 [ mn ( 6 m ( n ( ]. 2 Proof. Using the above Tables [ 7-2 ] an quation( we have 2 χ(g =. i i2 i3 i v i v i2 v i3 v i Since each path repeate twice therefore, we take 2 [ a a A i2 i3 i b i2 i3 i e b i2 i3 i c e i2 i3 i f c i2 i3 i f i2 i3 i ] =

14 Page 3 of 23 anaian Journal of hemistry 2 [( 2 Molecular topological properties of iamon like networks ( ( ( ( ( [( ( ] = n n ( 2n 2 2 ( n 8 ( n n 8 ( 2m 30m 8n36 2n 2 ( 2 2 2n 2 2mn 28m 2n6 ]. After simplification, we get = 2 [ mn ( 6 m ( n ( ]. 2 n n 2 2 2n 6 2 Theorem Let 3 X( ( L 2n L 2m be the graph, then thir sum connectivity of the graph is given by 58 2 [6mn m ( n ( ]. Proof. Using the above Tables [ 7-2 ] an quation(7 we have 2 χ(g =. i i2 i3 i v i v i2 v i3 v i Since each path repeate twice therefore we take 2 [ a a A i2 i3 i b b i2 i3 i c i2 i3 i e 2 [( 2 e i2 i3 i f c i2 i3 i f i2 i3 i ] = ( ( ( ( ( [( ] = 2n n ( 2n 22 ( n 8 ( n n 8 ( 2m 30m 8n36 2n ( 22 2n 2 2mn 28m 2n6 ]. After simplification, we get = 58 2 [6mn m ( n ( ]. n n 2 2 2n 6 2 Main results of extene Aztec iamon An Aztec iamon of orer n containing all squares lattice whose center (x, y satisfy x y n. Here n is fixe an square lattice contains unit squares with the origin as vertex of them, so that both p an q are half integer. An extene Aztec iamon is enote by AZ(n. An extene Aztec iamon is gaine by joining all vertices of egree 2. The subsequent graph is non-isomorphic to the generalize Aztec iamon having vertices of egree 3 also. There are three types of partite set in the ege partition of ege set of AZ(n. We compute thir connectivity an thir sum-connectivity inices of extene Aztec iamon.

15 anaian Journal of hemistry Page of 23 M. Imran, A. Q. aig, H. M. A. Siiqui, R. Sarwar AZ( AZ(2 A AZ(3 G H AZ(8 ig. 7: xtene Aztec iamon of ifferent imension. A A A A ig. 8: Types of three eges paths of extene Aztec iamon AZ(n that start from set A.

16 Page 5 of 23 anaian Journal of hemistry Molecular topological properties of iamon like networks 5 Three eges path connectivity is obtaine by twice the set A or 3 χ( ( AZ(n = 2A. Table 3: ge partition of ege set A of AZ(n, where n 8. ijkl arinality of ijkl ig. 9: Types of three eges paths of extene Aztec iamon AZ(n that start from set. Similarly, 3 χ ( AZ(n =2. Table : ge partition of ege set of AZ(n, where n 8. ijkl arinality of ijkl ig. 20: Types of three eges paths of extene Aztec iamon AZ(n that start from set. Similarly, 3 χ ( AZ(n = 2. 3 χ( ( AZ(n = 2. 3 χ ( AZ(n = (n. 3 χ ( AZ(n = 2. 3 χ ( AZ(n = 2G.

17 anaian Journal of hemistry Page 6 of 23 6 M. Imran, A. Q. aig, H. M. A. Siiqui, R. Sarwar Table 5: ge partition of ege set of AZ(n, where n 8. ijkl 33 arinality of ijkl 6 3 ig. 2: Types of three eges paths of extene Aztec iamon AZ(n that start from set. Table 6: ge partition of ege set of AZ(n, where n 8. ijkl 3 arinality of ijkl ig. 22: Types of three eges paths of extene Aztec iamon AZ(n that start from set. Table 7: ge partition of ege set of AZ(n, where n 8. ijkl 33 3 arinality of ijkl 8 6n 2 20n ig. 23: Types of three eges paths of extene Aztec iamon AZ(n that start from set. G G G G G ig. 2: Types of three eges paths of extene Aztec iamon AZ(n that start from set G.

18 Page 7 of 23 anaian Journal of hemistry Molecular topological properties of iamon like networks 7 Table 8: ge partition of ege set of AZ(n, where n 8. ijkl arinality of ijkl 3n 30 Table 9: ge partition of ege set G of AZ(n, where n 8. ijkl arinality of ijkl 3n 30 H H H H 33 3 ig. 25: Types of three eges paths of extene Aztec iamon AZ(n that start from set H. Table 20: ge partition of ege set H of AZ(n, where n 8. ijkl 33 3 arinality of ijkl 3n 30 ollowing theorems present analytically close formula of thir connectivity an thir sum-connectivity extene Aztec iamon. Theorem 5 Let 3 χ(az(n be the graph, then thir connectivity of the graph is given by 2 [n2 ( 8 8 n ( ]. Proof. Using the above Tables [ 3-9 ] an quation ( we have 2 χ(g =. i i2 i3 i v i v i2 v i3 v i Since each path repeate twice therefore, we take 2 [ a a A i2 i3 i b i2 i3 i e g i2 i3 i h H 2 [( 333 b i2 i3 i c e i2 i3 i f h i2 i3 i ] = c i2 i3 i f i2 i3 i g G ( ( ( 3 3 ( ( ] = ( ( [( ( ( ( 50 3 ( n2 20n 66 2 ( n 30 ( n 30 ( 3 3 3

19 anaian Journal of hemistry Page 8 of 23 8 M. Imran, A. Q. aig, H. M. A. Siiqui, R. Sarwar 3n 30 ]. After simplification, we get = 2 [n2 ( 8 8 n ( ]. Theorem 6 Let 3 X(AZ(n be the graph, then thir sum connectivity of the graph is given by 2 [n2 ( 2 n 6 3 n ( ]. Proof. Using the above Tables [ 3-9 ] an quation (7 we have 3 χ(g =. i i2 i3 i v i v i2 v i3 v i Since each path repeate twice therefore, we take 2 [ a a A i2 i3 i b i2 i3 i e g i2 i3 i h G 2 [( 333 b i2 i3 i c e i2 i3 i f h i2 i3 i ] = c i2 i3 i f i2 i3 i g G ( ( ( 3 3 ( ( ] = ( ( [( ( ( 33 6 ( 50 3 ( n2 20n 66 2 ( n 30 ]. 3n 30 ( After simplification, we get = 2 [n2 ( 2 n 6 3 n ( ]. 3n 30 ( 33 3 The graphical representation of thir connectivity an thir sum-connectivity for o generalize Aztec iamon is epicte in igure 26. The graphical representation of thir connectivity an thir sumconnectivity for even generalize Aztec iamon is epicte in igure 27. y increasing the values of m an n, the value of inices increases. The graphical representation of thir connectivity an thir sum-connectivity for extene Aztec iamon is epicte in igure 28. y increasing the values of n, the value of inices increase an thir sum connectivity inex approaches towars the thir connectivity inex. The comparison of thir connectivity an sum thir connectivity inex of both even an ol generalize Aztec iamon is epicte in igure 29 an 30 respectively. The graphical representation shows that the thir an sum thir connectivity inex of generalize Aztec iamon for even an o cases respectively are very close to each other.

20 Page 9 of 23 anaian Journal of hemistry Molecular topological properties of iamon like networks 9 ig. 26: The comparison of thir connectivity an thir sum-connectivity for o generalize Aztec iamon, thir connectivity inex Green, thir sum connectivity inex lue. ig. 27: The comparison of thir connectivity an thir sum-connectivity for even generalize Aztec iamon, thir connectivity inex Re an thir sum connectivity inex Green. Inex n ig. 28: The comparison of thir connectivity an thir sum-connectivity for extene Aztec iamon, thir connectivity inex Re an thir sum connectivity inex Green. ig. 29: The comparison of thir connectivity of generalize Aztec iamon for both even an o cases, thir connectivity inex for o lue an thir connectivity inex for even Green.

21 anaian Journal of hemistry Page 20 of M. Imran, A. Q. aig, H. M. A. Siiqui, R. Sarwar ig. 30: The comparison of sum thir connectivity inex of generalize Aztec iamon for both even an o cases, sum thir connectivity inex for o Re an thir sum connectivity inex for even case Green. 5 onclusion an general remarks In this paper, we consiere the conclusion thir connectivity an thir sum-connectivity for generalize Aztec iamon an extene iamon. We erive the close formulas of thir connectivity an thir sum-connectivity for generalize Aztec iamon an extene iamon. General connectivity an general sum-connectivity for these iamons can be consiere for future stuy. References. Ashrafi, A. R.; Nikzar, P.; igest J. Nanomater. iostructures. 2009,, Ashrafi, A. R.; Nikza, P.; J. Nanomater. ios., 2009,, aig, A. Q.; Imran, M.; Ali, H.; anaian Journal of chemistry, 205, 93, ollobas,.; ros, P.; Arts ombin., 998, 50, evillers, J.; alabn A. T.; Goron an reach, Amsteram., 2000, 3, iuea, M. V.; Jhon, P..; MATH ommun. Math. omput. hem., 200,, iuea,m. V.; Graovac, A.; MATH ommun. Math. omput. hem., 200,, iuea, M. V.; Hosoya; MATH ommun. Math. omput. hem., 2002, 5, u, Z.; Zhou,.; Trinajstić, N.; J. Math. hem., 200, 7, u, Z.; Zhou,.; Sep 2, (2009 arxiv: v.. u, Z.; Zhou,.; Trinajstić, N.; Appl. Math. Lett., 200, 2, straa,.; J. hem. Inf. omput. Sci., 995, 35, straa,.; hem. Phys. Lett., 999, 32, u, S. P.; u, T. S.; The lectronic J. ombin., 2005, 2, # R8. 5. arahani, M. R.; Acta him. Slov., 202, 59, arahani, M. R.; Kato, K.; Vla. M. P.; Stuia U. hemia, 203, 58, Gutman, I.; urtula,.; Math. hem. Monogr. 6, Univ. Kragujevac, Kragujevac Li, X.; Gutman, I.; Math. hem. Monogr. I, Univ. Kragujevac, Kragujevac Hayat. S.;, Malik, M. A.; Imran, M.; Romanian journal of Information science an technology 205, 8,.

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23 anaian Journal of hemistry Page 22 of M. Imran, A. Q. aig, H. M. A. Siiqui, R. Sarwar igure aptions ig.: Aztec iamon of ifferent imensions. ig.2: Generalize Aztec iamon L L. ig.3: Types of three eges path connectivity of o generalize Aztec iamon L L. ig.: Types of three eges paths of o generalize Aztec iamon L L that start from set A. ig.5: Types of three eges paths of o generalize Aztec iamon L L that start from set. ig.6: Types of three eges paths of o generalize Aztec iamon L L that start from set. ig.7: Types of three eges paths of o generalize Aztec iamon L L that start from set. ig.8: Types of three eges paths of o generalize Aztec iamon L L that start from set. ig.9: Types of three eges paths of o generalize Aztec iamon L L that start from set. ig.0: Types of three eges path of even generalize Aztec iamon L L. ig.: Types of three eges paths of even generalize Aztec iamon L L that start from set A. ig.2: Types of three eges paths of even generalize Aztec iamon, that start from set. ig.3: Types of three eges paths of even generalize Aztec iamon L L that start from set. ig.: Types of three eges paths of even generalize Aztec iamon L L that start from set. ig.5: Types of three eges paths of even generalize Aztec iamon that start from set. ig.6: Types of three eges paths of even generalize Aztec iamon that start from set. ig.7: xtene Aztec iamon of ifferent imension. ig.8: Types of three eges paths of extene Aztec iamon AZ(n that start from set A. ig.9: Types of three eges paths of extene Aztec iamon AZ(n that start from set. ig.20: Types of three eges paths of extene Aztec iamon AZ(n that start from set. ig.2: Types of three eges paths of extene Aztec iamon AZ(n that start from set. ig.22: Types of three eges paths of extene Aztec iamon AZ(n that start from set. ig.23: Types of three eges paths of extene Aztec iamon AZ(n that start from set. ig.2: Types of three eges paths of extene Aztec iamon AZ(n that start from set G. ig.25: Types of three eges paths of extene Aztec iamon AZ(n that start from set H. ig.26: The comparison of thir connectivity an thir sum-connectivity for o generalize Aztec iamon, thir connectivity inex Green, thir sum connectivity inex lue. ig.27: The comparison of thir connectivity an thir sum-connectivity for even generalize Aztec iamon, thir connectivity inex Re an thir sum connectivity inex Green. ig.28: The comparison of thir connectivity an thir sum-connectivity for extene Aztec iamon, thir connectivity inex Re an thir sum connectivity inex Green. ig.29: The comparison of thir connectivity of generalize Aztec iamon for both even an o cases, thir connectivity inex for o lue an thir connectivity inex for even Green. ig.30: The comparison of sum thir connectivity inex of generalize Aztec iamon for both even an

24 Page 23 of 23 anaian Journal of hemistry Molecular topological properties of iamon like networks 23 o cases, sum thir connectivity inex for o Re an thir sum connectivity inex for even case Green.