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GeoloGy, ecoloGy, and landscapes, 2017 Vol . 1, no . 3, 173– 183 https://doi.org/10.1080/24749508.2017.1361143 INWASCON OPEN ACCESS a b c d Wei Gao , M. R. Rajesh Kanna , E. Suresh and Mohammad Reza Farahani a b s chool of Information s cience and Technology, yunnan normal University, Kunming, china; d epartment of Mathematics, Maharani’s c d s cience c ollege for Women, Mysore, India; d epartment of Mathematics, Velammal engineering c ollege, chennai, India; d epartment of applied Mathematics, Iran University of s cience and Technology, Tehran, Iran ABSTRACT ARTICLE HISTORY A larger amount of studies reveal that there is strong inherent connection between the chemical Received 19 april 2017 a ccepted 21 July 2017 characteristics of nanostructures and their molecular structures. Degree-based topological indices introduced on these chemical molecular structures can help material scientists better KEYWORDS understand its chemical and biological features, thus they can make up for the lack of chemical Theoretical chemistry; experiments. In this paper, by means of edge dividing trick, we present several degree-based general sum connectivity indices of special widely employed nanostructures: SC C [p, q] nanotubes, polyphenylene index; general Randic index; 5 7 H-naphtalenic nanotube; dendrimers, H-Naphtalenic nanotubes NPHX[m, n], TUC [m,n] nanotubes and PAMAM pa Ma M dendrimer dendrimers. 1. Introduction Farahani, 2016; Gao, Wang, & Farahani, 2016) for more detail). The notations and terminologies used but not As the development of nanotechnology, more and more clearly defined in our article can be referred in book of nanomaterials are emerging every year. Thus, identifi- (Bondy & Mutry, 2008) written by Bondy and Mutry. cation of the chemical properties of these nanomaterials Bollobas and Erdos (1998) defined the general Randic has become more and more cumbersome. Fortunately, index which was stated as follows: previous studies have shown that chemical character- istics of nanomaterials and their molecular structures R (G)= (d(u)d(v)) , (1) are closely related. By defining the chemical topological uv∈E(G) indices to study indicators of these nanostructures can where k is a real number and d(u) denotes the degree of help researchers to determine their chemical properties, vertex u in molecular graph G. Liu and Gutman (2007) which make up the chemical experiments defects. determined the estimating for general Randic index and Specifically, the nanostructure is modelled as a graph, its special cases. Throughout, we always assume that k where each vertex represents an atom and each edge is a real number. denotes a chemical bond between two atoms. Let G be By taking k = 1 and k = −1, formula (1) then becomes a (molecular) graph with vertex set V(G) and edge set the second Zagreb index (M (G)) and the modified sec- E(G). A topological index can be regarded as a real-val- ond Zagreb index (M (G)), respectively: ued function f: G→ ℝ which maps each nanostructure to a real number. As numerical descriptors of the molec- M (G)= d(u)d(v), M (G)= . ular structure yielded from the corresponding nanos- 2 2 d(u)d(v) uv∈E(G) uv∈E(G) tructures, topological indices have been proofed several applications in nanoengineering, for example, QSPR/ Zhou and Trinajstic (2010) introduced the general sum QSAR study. In the past years, harmonic index, Wiener connectivity as follows: index, sum connectivity index were introduced to meas- ure certain structural features of nanomolecules. There (G)= (d(u)+ d(v)) . (2) were several papers contributing to determine these top- uv∈E(G) ological indices of special molecular graph in chemical By taking k = , formula (2) becomes the sum connec- engineering (See Hosamani (2016), (Gao & Farahani, tivity index (χ(G)) which is formulated by: 2016; Gao & Wang, 2014, 2015, 2016, 2017), Gao and Farahani (2016), and (Gao, Farahani, & Jamil, 2016; Gao, (G)= (d(u)+ d(v)) . uv∈E(G) Farahani, & Shi, 2016; Gao, Siddiqui, Imran, Jamil, & CONTACT Wei Gao gaowei@ynnu.edu.cn © 2017 The a uthor(s). published by Informa UK limited, trading as Taylor & Francis Group. This is an open a ccess article distributed under the terms of the creative c ommons a ttribution license (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 174 W. GAO ET AL. Gao and Wang (2016) introduced the general harmonic Several conclusions on PM (G) and PM (G)can be 1 2 index as: referred to Eliasi, Iranmanesh, and Gutma (2012) and Xu and Das (2012). Furthermore, Ranjini, Lokesha, and Usha (2013) H (G)= . (3) d(u) + d(v) re-defined the Zagreb indices, i.e., the redefined first, uv∈E(G) second and third Zagreb indices of a (molecular) graph If we take k = 1 in formula (3), then it becomes a normal G were manifested as follows: harmonic index which was described by: d(u)+ d(v) ReZG (G)= , d(u)d(v) e=uv∈E(G) H(G)= . d(u) + d(v) uv∈E(G) d(u)d(v) Eliasi and Iranmanesh (2011) reported the ordinary ReZG (G)= d(u)+ d(v) geometric–arithmetic index (or, called general geomet- e=uv∈E(G) ric–arithmetic index) as the extension of geomet- ric–arithmetic index which was stated as follows: and � � ReZG (G)= d(u)d(v)(d(u) + d(v)), 2 d(u)d(v) OGA (G)= . e=uv∈E(G) d(u) + d(v) uv∈E(G) respectively. Clearly, GA (geometric–arithmetic) index is a special Although there have been several advances in dis- case of ordinary geometric–arithmetic index when k = 1. tance-based indices of molecular graphs, the study of Azari and Iranmanesh (2011) proposed the general- degree-based indices for special nanomolecular struc- ized Zagreb index of molecular graph G expressed by: tures are still largely limited. In addition, as widespread and critical nanostructures, SC C [p, q] nanotubes, 5 7 t t t t 1 2 2 1 M (G)= d(u) d(v) + d(u) d(v) , {t ,t } 1 2 polyphenylene dendrimers, H-Naphtalenic nanotubes uv∈E(G) NPHX[m, n], TUC [m, n] nanotubes and PAMAM den- where t and t are arbitrary non-negative integers. drimers are widely used in medical science and material 1 2 Several polynomials related to degree-based indices field. For these reasons, we give the exact expressions are also introduced. For instance, the first and the second of above-mentioned degree-based indices for these Zagreb polynomials are expressed by: nanostructures. e Th rest of the context is arranged as follows: first, d(u)+d(v) M (G, x)= x we present the degree-based indices of SC C p, q 5 7 uv∈E(G) nanotubes; then, the nanostructure of polyphenylene and dendrimers are considered; third, we focus on the H-Naphtalenic nanotubes NPHX[m, n]; the degree- d(u)d(v) M (G, x)= x , based indices computation of TUC [m, n] nanotubes are uv∈E(G) presented in Section 5; at last, we consider three kinds of respectively. PAMAM dendrimers: PD [n], PD [n] and DS [n]. 1 2 1 Moreover, the third Zagreb index and third Zagreb polynomial are denoted as: 2. Degree-based indices of SC C [p, q] 5 7 nanotubes M (G)= d(u) − d(v) uv∈E(G) e p Th urpose of this section is to manifest several degree- based indices of SC C [p, q] nanotubes. Actually, this and 5 7 nanotube is a kind of C C -net which is obtained by 5 7 d(u)−d(v) M (G, x)= x . alternating C and C . This classical tiling of C and C 5 7 5 7 uv∈E(G) can either cover a cylinder or a torus. A period of SC C 5 7 [p, q] (here p is the number of heptagons in each row e m Th ultiplicative version of first and second Zagreb and q is the number of periods in whole lattice) is con- indices were introduced by Gutman (2011) and sisted of three rows (see Figure 1 for more details on i-th Ghorbani and Azimi (2012) as follows: period). Clearly, there are 8p vertices in one period of PM (G)= (d(u) + d(v)), the lattice, and thus V SC C p, q = 8pq. Using the 5 7 e=uv∈E(G) similar fashion, there are 12p edges in one period and and exists 2p extra edges joined to the end of this nanostruc- PM (G)= (d(u)d(v)). 2 ture. Therefore, we have E SC C p, q = 12pq − 2p. 5 7 e=uv∈E(G) GEOLOGY, ECOLOGY, AND LANDSCAPES 175 Figure 1. i-th period of SC C [p, q] nanotube. 5 7 e m Th ain technique in this paper to obtain the desired ReZG SC C p, q = 648pq − 290p. conclusion is edge dividing approach. Throughout this 3 5 7 paper, we use the following notations for edge dividing. Let δ(G) and Δ(G) be the minimum and maxi- Proof: By observation of nanostructure SC C p, q , 5 7 mum degree of G. We divide edge set E(G) and ver- we infer three partitions of edge set: tex set V(G) into several partitions: for any i, 2δ(G) ≤ i ≤ 2Δ(G), let E = {e=uv∊E(G)|d(v)+d(u)=i}; for any j, • E (or E ): d(u) = d(v) = 2; i 4 4 2 2 ∗ (δ(G)) ≤ j ≤ (Δ(G)) , let E = {e=uv∊E(G)|d(v) d(u) = j} • E (or E ), d(u) = d(v) = 3; 6 9 and for any k, δ(G) ≤ k ≤ Δ(G), let V = {v ∊ V(G) | • E (or E ), d(u) = 2 and d(v) = 3. k 5 6 d(v) = k}. ∗ ∗ Furthermore, we get E = E = p , E = E = 6p , 4 4 5 6 Now, we state the main conclusion in this section. and E = E = 12pq − 9p. Then, the result follows 6 9 Theorem 1: from the definitions of these degree-based indices. k k k R SC C p, q = 12pq − 9p ⋅ 9 + 6p ⋅ 6 + p ⋅ 4 , k 5 7 Remark 1: From what we have deduced in e Th orem k k k 1, we yield that SC C p, q = 12pq − 9p ⋅ 6 + 6p ⋅ 5 + p ⋅ 4 , k 5 7 M SC C p, q = pq + , 2 5 7 3 4 H SC C p, q = 12pq − 9p ⋅ k 5 7 k k 2 1 + 6p ⋅ + p ⋅ , H SC C p, q = 4pq − p, 5 7 5 2 � � M SC C p, q = 6p. 3 5 7 � � �� 2 6 OGA SC C p, q = (12pq − 8p)+ 6p ⋅ , k 5 7 3. Degree-based indices of polyphenylene t +t 1 2 M SC C p, q = 24pq − 18p ⋅ 3 {t ,t } 5 7 1 2 dendrimers t t t t t +t +1 1 2 2 1 1 2 + 6p ⋅ 2 3 + 2 3 + p ⋅ 2 , e a Th im of this section is to show the degree-based indices of polyphenylene dendrimers D [n] and D [n], 4 2 6 5 4 M SC C p, q , x = 12pq − 9p x + 6px + px , where n ∊ ℕ. These two molecular structures are widely 1 5 7 appeared in the nanomaterials. The kernel structure of D [n] and D [n] can be referred to Figure 2. 4 2 9 6 4 M SC C p, q , x = 12pq − 9p x + 6px + px , Additionally, the following Figure 3 present the D [n] 2 5 7 with three growth stages. e m Th ain results in this section are manifested as M SC C p, q , x = 6px + 12pq − 8p , follows: 3 5 7 k n k Theorem 2: R (D [n]) = 4 ⋅ 12 + (36 ⋅ 2 − 36) ⋅ 9 k 4 n k + (48 ⋅ 2 − 40) ⋅ 6 + (56 ⋅ 2 12pq−9p 6p p PM SC C p, q = 6 5 4 , n k 1 5 7 − 40) ⋅ 4 , k n k n χ (D [n]) = 4 ⋅ 7 + (36 ⋅ 2 − 36) ⋅ 6 + (48 ⋅ 2 − 40) ⋅ k 4 k n k 5 + (56 ⋅ 2 − 40) ⋅ 4 , 12pq−9p 6p p PM SC C p, q = 9 6 4 , 2 5 7 k k 2 1 H D [n] = 4 ⋅ + (36 ⋅ 2 − 36) ⋅ k 4 7 3 ReZG SC C p, q = 8pq, 1 5 7 + (48 ⋅ 2 − 40) ⋅ + (56 ⋅ 2 − 40) ⋅ , ReZG SC C p, q = 18pq − p, 2 2 5 7 10 176 W. GAO ET AL. Figure 2. The kernel of D [n] and D [n], respectively. 4 2 Figure 3. polyphenylene dendrimers D [n] with three growth stages. � � � � √ √ k k � � 4 3 2 6 12 n 9 M D [n], x = 4x + (36 ⋅ 2 − 36)x OGA D [n] = 4 ⋅ + (48 ⋅ 2 − 40) ⋅ 2 4 k 4 7 5 n 6 n 4 + (48 ⋅ 2 − 40)x + (56 ⋅ 2 − 40)x , +(92 ⋅ 2 − 76), n n t t t t M D [n], x = (48 ⋅ 2 − 36)x + (92 ⋅ 2 − 76), 1 2 2 1 3 4 M D [n] = 4 ⋅ 3 4 + 4 3 {t ,t } 4 1 2 n t +t 1 2 + (72 ⋅ 2 − 72) ⋅ 3 n n n 36⋅2 −36 48⋅2 −40 56⋅2 −40 n t t t t PM D [n] = 2401 ⋅ 6 5 4 , 1 2 2 1 1 4 + 48 ⋅ 2 − 40 ⋅ 2 3 + 2 3 ( ) n t +t +1 1 2 + 56 ⋅ 2 − 40) ⋅ 2 , n n n 36⋅2 −36 48⋅2 −40 56⋅2 −40 7 n 6 PM D [n] = 20736 ⋅ 9 6 4 , 2 4 M D [n], x = 4x + (36 ⋅ 2 − 36)x 1 4 n 5 n 4 + (48 ⋅ 2 − 40)x + (56 ⋅ 2 − 40)x , ReZG D [n] = 120 ⋅ 2 − 95, 1 4 GEOLOGY, ECOLOGY, AND LANDSCAPES 177 n t +t 1 2 M D [n] = (72 ⋅ 2 − 70) ⋅ 3 {t ,t } 2 1 2 838 946 ReZG D [n] = ⋅ 2 − , n t t t t 2 4 1 2 2 1 + (48 ⋅ 2 − 44) ⋅ 2 3 + 2 3 5 7 n t +t +1 1 2 + (56 ⋅ 2 − 48) ⋅ 2 , ReZG D [n] = 4280 ⋅ 2 − 3448. 3 4 n 6 n 5 Proof: By observation of polyphenylene dendrimers M D [n], x = (36 ⋅ 2 − 35)x + (48 ⋅ 2 − 44)x 1 2 D [n], we infer four partitions of edge set: n 4 + (56 ⋅ 2 − 48)x , • E (or E ): d(u) = d(v) = 2; n 9 n 6 4 M D [n], x = (36 ⋅ 2 − 35)x + (48 ⋅ 2 − 44)x 2 2 • E (or E ), d(u) = d(v) = 3; 9 n 4 + (56 ⋅ 2 − 48)x , • E (or E ), d(u) = 2 and d(v) = 3; • E (or E ), d(u) = 3 and d(v) = 4. n n M D [n], x = (48 ⋅ 2 − 44)x + (92 ⋅ 2 − 83), 3 2 ∗ n Furthermore, we get E = E = 56 ⋅ 2 − 40, 4 4 ∗ n ∗ n E = E = 48 ⋅ 2 − 40, E = E = 36 ⋅ 2 − 36 , 5 6 6 9 n n n 36⋅2 −35 48⋅2 −44 56⋅2 −48 and E = E = 4. Then, the result follows from the PM D [n] = 6 5 4 , 7 12 1 2 definitions of these degree-based indices. ✷ Remark 2: From what we have obtained in e Th orem n n n 36⋅2 −35 48⋅2 −44 56⋅2 −48 PM D [n] = 9 6 4 , 2, we yield that 2 2 M D [n] = 680 ⋅ 2 − 548, 1 4 ReZG D [n] = 120 ⋅ 2 − 108, 1 2 M D [n] = 836 ⋅ 2 − 676, 2 4 838 1533 ∗ n ReZG D [n] = ⋅ 2 − , 2 2 M D [n] = 26 ⋅ 2 − , 2 4 5 10 n n � � 4 36 ⋅ 2 − 36 48 ⋅ 2 − 40 D [n] = + + ReZG D [n] = 4280 ⋅ 2 − 3978. √ √ √ 4 3 2 7 6 5 + 28 ⋅ 2 − 20, Proof: By observation of polyphenylene dendrimers D [n], we get three partitions of edge set: H D [n] = ⋅ 2 − 47, • E (or E ): d(u) = d(v) = 2; • E (or E ), d(u) = d(v) = 3; √ √ 9 � � 16 3 6 • E (or E ), d(u) = 2 and d(v) = 3. n n GA D [n] = + (96 ⋅ 2 − 80) + (92 ⋅ 2 − 76), 7 5 ∗ n Additionally, we have E = E = 56 ⋅ 2 − 48, 4 4 ∗ n ∗ n E = E = 48 ⋅ 2 − 44, and E = E = 36 ⋅ 2 − 35 . 5 6 6 9 M D [n] = 48 ⋅ 2 − 36. 3 4 u Th s, the result follows from the definitions of these degree- based indices. ✷ n k Theorem 3: R (D [n]) = (36 ⋅ 2 − 35) ⋅ 9 + (48 k 2 Remark 3: From what we have obtained in Theorem n k n ⋅ 2 − 44) ⋅ 6 + (56 ⋅ 2 3, we yield that − 48) ⋅ 4 , M D [n] = 680 ⋅ 2 − 622, 1 2 n k n k D [n] = (36 ⋅ 2 − 35) ⋅ 6 + (48 ⋅ 2 − 44) ⋅ 5 k 2 n k + (56 ⋅ 2 − 48) ⋅ 4 , n M D [n] = 836 ⋅ 2 − 771, 2 2 k k 1 2 n n H D [n] = (36 ⋅ 2 − 35) ⋅ + (48 ⋅ 2 − 44) ⋅ 209 ∗ n k 2 M D [n] = 26 ⋅ 2 − , 3 5 2 2 + (56 ⋅ 2 − 48) ⋅ , n n � � � √ � k 36 ⋅ 2 − 35 48 ⋅ 2 − 44 D [n] = + + 28 ⋅ 2 − 24, √ √ � � 2 6 2 OGA D [n] = (48 ⋅ 2 − 44) ⋅ 6 5 k 2 296 799 + (92 ⋅ 2 − 83), n H D [n] = ⋅ 2 − , 5 15 178 W. GAO ET AL. � √ � � � n n 2 6 GA D [n] = (96 ⋅ 2 − 88) + (92 ⋅ 2 − 83), OGA (NPHX[m, n]) = 15mn − 10m + 8m ⋅ , 5 k n t +t 1 2 M D [n] = 48 ⋅ 2 − 44. M (NPHX[m, n]) = (30mn − 20m) ⋅ 3 3 2 {t ,t } 1 2 t t t t 1 2 2 1 4. Degree-based indices of H-Naphtalenic + 8m ⋅ 2 3 + 2 3 , nanotubes 6 5 M NPHX m, n , x = 15mn − 10m x + 8mx , ( [ ] ) ( ) In this part, we consider the degree-based indices of 9 6 H-Naphtalenic nanotubes NPHX[m, n] (here m is M (NPHX[m, n], x) = (15mn − 10m)x + 8mx , denoted as the number of pairs of hexagons in first row and n is represented as the number of alternative hexa- M (NPHX[m, n], x) = 8mx + (15mn − 10m), gons in a column) which is a trivalent decoration with 15mn−10m 8m PM (NPHX[m,n]) = 6 5 , sequence of C , C , C , C , C , C , … in the first row and 6 6 4 6 6 4 a sequence of C , C , C , C , … in the other rows. That is 6 8 6 8 15mn−10m 8m to say, this nanolattice can be regarded as a plane tiling PM NPHX m,n = 9 6 , ( [ ]) of C , C and C . Thus, such type of tiling can either 4 6 8 cover a cylinder or a torus (see Figure 4 as an example). ReZG (NPHX[m, n]) = 10mn, Moreover, we can verify that V (NPHX[m, n]) = 10mn 45 27 and E(NPHX[m, n]) = 15mn − 2m. ReZG (NPHX[m, n]) = mn − m, 2 5 Now, we present the main results in this section. k Theorem 4: R (NPHX[m, n]) = (15mn − 10m) ⋅ 9 ReZG (NPHX[m, n]) = 810mn − 300m. + 8m ⋅ 6 , Proof: By observation of H-Naphtalenic nanotubes k k NPHX[m, n], we know two partitions of edge set: (NPHX[m, n]) = (15mn − 10m) ⋅ 6 + 8m ⋅ 5 , • E (or E ), d(u) = d(v) = 3; k k • E (or E ), d(u) = 2 and d(v) = 3. 1 2 H (NPHX[m, n]) = (15mn − 10m) ⋅ + 8m ⋅ , 3 5 Figure 4. The molecular structure of NPHX [n, n]. GEOLOGY, ECOLOGY, AND LANDSCAPES 179 k k Moreover, it is not hard to check that E = E = 8m 1 2 5 6 H TUC [m, n] = 2m ⋅ + 2m ⋅ k 4 and E = E = 15mn − 10m. Thus, we get the desired 3 7 6 9 formulations in terms of the definitions of these degree- + m(2n − 3) ⋅ , based indices. ✷ Remark 4: Using the conclusions obtained in Theorem � � 4, we yield that � � 4 3 OGA TUC [m, n] = 2mn − m + 2m ⋅ , k 4 M (NPHX[m,n]) = 90mn − 20m, t +t t t t t M (NPHX[m,n]) = 135mn − 42m, 1 2 1 2 2 1 2 M TUC [m, ] = 4m ⋅ 3 + 2m ⋅ 3 4 + 3 4 {t ,t } 4 1 2 2t +2t +1 1 2 5 2 ∗ + m(2n − 3) ⋅ 2 , M (NPHX[m,n]) = mn + m, 3 9 6 7 8 M TUC [m,n],x = 2mx + 2mx + m(2n − 3)x , 1 4 H(NPHX[m, n]) = 5mn − m, 9 12 16 M TUC [m,n],x = 2mx + 2mx + m(2n − 3)x , 2 4 M (NPHX[m,n]) = 8m. M TUC [m,n],x = 2mx + (2mn − m), 3 4 2m 2m m(2n−3) 5. Degree-based indices of TUC [m, n] PM TUC [m,n] = 6 7 8 , 1 4 nanotubes 2m 2m m(2n−3) PM TUC [m,n] = 9 12 16 , In this part, we discuss the degree-based indices of 2 4 TUC [m, n] nanotubes (here m is denoted as the number ReZG TUC [m, n] = mn − m, of squares in a row and n is represented as the number 1 4 of squares in a column) which is a plane tiling of C . This tessellation of C can either cover a cylinder or a ReZG TUC [m, n] = 4mn + m, 2 4 torus. We verify that V TUC [m,n] = m (n + 1) and E TUC [m, n] = 2 mn + m. Figure 5 describes the 3D ReZG TUC [m, n] = 256mn − 108m. 3 4 representation of this kind of nanostructure. Again, using the trick of edge dividing, we get the Proof: By observation of TUC [m, n] nanotubes, following statement. we ensure that its edge set can be divided into three k k partitions: Theorem 5: R TUC [m, n] = 2m ⋅ 9 + 2m ⋅ 12 +m k 4 × (2n − 3) ⋅ 16 , • E (or E ), d(u) = d(v) = 3; k k k ∗ TUC [m, n] = 2m ⋅ 6 + 2m ⋅ 7 + m(2n − 3) ⋅ 8 , • E (or E ), d(u) = 3 and d(v) = 4; k 4 12 • E (or E ), d(u) = d(v) = 4. Moreover, it is not hard to check that E = E = 2m , 6 9 ∗ ∗ E = E = 2m and E = E = m(2n − 3). 7 12 8 16 er Th efore, we obtain the desired formulations in terms of the definitions of these degree-based indices. Remark 5: According to results presented in Theorem 5, we have M TUC [m,n] = 16mn + 2m, 1 4 M TUC [m,n] = 32mn − 6m, 2 4 1 13 M TUC [m,n] = mn + m, 2 4 8 144 1 41 H TUC [m,n] = mn + m, 2 84 Figure 5. The 3d expression of TUC [6, n]. M TUC [m,n] = 2m. 3 4 180 W. GAO ET AL. • E (or E ), d(u) = 1 and d(v) = 2; 6. Degree-based indices of PAMAM 3 2 • E , d(u) = 1 and d(v) = 3; dendrimers • E , d(u) = d(v) = 2; In this section, we first discuss the degree-based indi- • E (or E ), d(u) = 2 and d(v) = 3. 5 6 ces of PAMAM dendrimers with trifunctional core unit constructed by dendrimer generations G with n growth Moreover, it is not hard to check that ∗ n ∗ n ∗ n stages. We use PD to denote this nanostructures with E = E = 3 ⋅ 2 , E = 6 ⋅ 2 − 3, E = 18 ⋅ 2 − 9 1 3 2 3 4 ∗ n n growth stages. and E = E = 21 ⋅ 2 − 12. At last, the results 5 6 n k n k obtained by means of definitions of these degree-based Theorem 6: R (PD ) = 3 ⋅ 2 ⋅ 2 + (6 ⋅ 2 − 3) ⋅ 3 + k 1 n k n k indices. ✷ (18 ⋅ 2 − 9) ⋅ 4 + (21 ⋅ 2 − 12) ⋅ 6 , Remark 6: By taking the special value of k in results n k n k PD = 3 ⋅ 2 ⋅ 3 + (24 ⋅ 2 − 12) ⋅ 4 k 1 of Theorem 5, we get n k + (21 ⋅ 2 − 12) ⋅ 5 , M PD = 210 ⋅ 2 − 108, 1 1 k k 2 1 n n H PD = (3 ⋅ 2 ) ⋅ + (24 ⋅ 2 − 12) ⋅ n k 1 M PD = 222 ⋅ 2 − 117, 3 2 2 1 + (21 ⋅ 2 − 12) ⋅ , 23 21 ∗ n 5 M PD = ⋅ 2 − , 2 1 2 4 � � � � √ √ k k n n � � 3 ⋅ 2 21 ⋅ 2 − 12 � � 2 2 3 PD = + 12 ⋅ 2 − 6 + , n n √ √ OGA PD = 3 ⋅ 2 + (6 ⋅ 2 − 3) ⋅ k 1 3 5 3 2 � � 2 6 n n 112 54 + (18 ⋅ 2 − 9) + (21 ⋅ 2 − 12) ⋅ , H PD = ⋅ 2 − , 5 5 � � n t t n t t 1 2 1 2 GA PD = 6 ⋅ 2 − 3 ⋅ ( ) M PD = 3 ⋅ 2 2 + 2 + (6 ⋅ 2 − 3) 3 + 3 {t ,t } 1 1 2 �� � � n t +t +1 1 2 + (18 ⋅ 2 − 9)2 n + 18 + 2 2 ⋅ 2 − 9 n t t t t 1 2 2 1 + (21 ⋅ 2 − 12) 2 3 + 2 3 , √ + (42 ⋅ 2 − 24) ⋅ , n 3 n 4 n 5 M PD , x = 3 ⋅ 2 x + (24 ⋅ 2 − 12)x + (21 ⋅ 2 − 12)x , 1 1 M PD = 36 ⋅ 2 − 18. 3 1 n 2 n 3 n 4 M PD , x = 3 ⋅ 2 x + (6 ⋅ 2 − 3)x + (18 ⋅ 2 − 9)x 2 1 Next, we determine the degree-based indices of PAMAM n 6 + (21 ⋅ 2 − 12)x , dendrimer with different core constructed by dendrimer generations G with n growth stages. We use PD to n 2 n 2 n denote this nanostructures with n growth stages. M PD , x = (6 ⋅ 2 − 3)x + (24 ⋅ 2 − 12)x 3 1 n k n k Theorem 7: R (PD ) = 4 ⋅ 2 ⋅ 2 + (8 ⋅ 2 − 4) ⋅ 3 + + (18 ⋅ 2 − 9), k 2 n k n (24 ⋅ 2 − 11) ⋅ 4 + (28 ⋅ 2 − n n n 3⋅2 24⋅2 −12 21⋅2 −12 PM PD = 3 4 5 , 14) ⋅ 6 , 1 1 n k n k n n n n PD = 4 ⋅ 2 ⋅ 3 + 32 ⋅ 2 − 15 ⋅ 4 3⋅2 6⋅2 −3 18⋅2 −9 21⋅2 −12 ( ) k 2 PM PD = 2 3 4 6 , 2 1 n k + (28 ⋅ 2 − 14) ⋅ 5 , ReZG PD = 48 ⋅ 2 − 23, k k 1 1 2 1 n n H PD = (4 ⋅ 2 ) ⋅ + (32 ⋅ 2 − 15) ⋅ k 2 3 2 ReZG PD = 42 ⋅ 2 − , 2 1 2 + (28 ⋅ 2 − 14) ⋅ , � � � � √ √ k k ReZG PD = 1008 ⋅ 2 − 540. 3 1 � � 2 2 3 n n OGA PD = 4 ⋅ 2 + (8 ⋅ 2 − 4) ⋅ k 2 3 2 Proof: By observation of PAMAM dendrimer PD , � � we ensure that its edge set can be divided into four 2 6 n n + (24 ⋅ 2 − 11) + (28 ⋅ 2 − 14) ⋅ , partitions: 5 GEOLOGY, ECOLOGY, AND LANDSCAPES 181 688 131 n t t n t t H PD = ⋅ 2 − , 1 2 1 2 M PD = 4 ⋅ 2 2 + 2 + 8 ⋅ 2 − 4 3 + 3 2 ( ) {t ,t } 2 15 10 1 2 n t +t +1 1 2 + (24 ⋅ 2 − 11)2 � � 8 2 n n n t t t t 1 2 2 1 GA PD = ⋅ 2 + (4 ⋅ 2 − 2) 3 +(28 ⋅ 2 − 14) 2 3 + 2 3 , 2 n 3 n 4 n n M PD , x = 4 ⋅ 2 x + (32 ⋅ 2 − 15)x + (24 ⋅ 2 − 11) + (56 ⋅ 2 − 28) ⋅ , 1 2 n 5 + (28 ⋅ 2 − 14)x , M PD = 48 ⋅ 2 − 22. 3 2 n 2 n 3 M PD , x = 4 ⋅ 2 x + (8 ⋅ 2 − 4)x 2 2 n 4 + 24 ⋅ 2 − 11 x ( ) At last, we compute the degree-based indices of other n 6 + (28 ⋅ 2 − 14)x , kinds of PAMAM dendrimer DS with n growth stages. n k n k Theorem 8: R (DS ) = (14 ⋅ 3 − 10) ⋅ 4 + (4 ⋅ 3 − 4) ⋅ 8 , k 1 n 2 n M PD , x = (8 ⋅ 2 − 4)x + (32 ⋅ 2 − 14)x 3 2 n k n k DS = 4 ⋅ 3 ⋅ 5 + 10 ⋅ 3 − 10 ⋅ 4 + (24 ⋅ 2 − 11), ( ) k 1 n k + (4 ⋅ 3 − 4) ⋅ 6 , n n n 4⋅2 32⋅2 −15 28⋅2 −14 PM PD = 3 4 5 , 1 2 k k 2 1 n n H DS = (4 ⋅ 3 ) ⋅ + (10 ⋅ 3 − 10) ⋅ k 1 n n n n 5 2 4⋅2 8⋅2 −4 24⋅2 −11 28⋅2 −14 PM PD = 2 3 4 6 , 2 2 k + (4 ⋅ 3 − 4) ⋅ , ReZG PD = 64 ⋅ 2 − 28, 1 2 � � � � n n OGA DS = 4 ⋅ 3 + (10 ⋅ 3 − 10) k 1 994 154 n � � ReZG PD = ⋅ 2 − , 2 2 2 2 15 5 + (4 ⋅ 3 − 4) ⋅ , ReZG PD = 1344 ⋅ 2 − 644. 3 2 n t t n t +t +1 1 2 1 2 Proof: By analysis of PAMAM dendrimer PD struc- M DS = 4 ⋅ 3 ⋅ 4 + 4 + (10 ⋅ 3 − 10)2 {t ,t } 1 1 2 tures, we yield the four dividings of its edge set. n t +2t 2t +t 1 2 1 2 + (4 ⋅ 3 − 4) 2 + 2 , • E (or E ), d(u) = 1 and d(v) = 2; n 5 n 4 n 6 • E , d(u) = 1 and d(v) = 3; 3 M DS , x = 4 ⋅ 3 x + (10 ⋅ 3 − 10)x + (4 ⋅ 3 − 4)x , 1 1 • E , d(u) = d(v) = 2; • E (or E ), d(u) = 2 and d(v) = 3. n 4 n 8 M DS , x = (14 ⋅ 3 − 10)x + (4 ⋅ 3 − 4)x , Moreover, it is not hard to check that 2 1 ∗ ∗ ∗ n n n E = E = 4 ⋅ 2 , E = 8 ⋅ 2 − 4, E = 24 ⋅ 2 − 11 3 2 3 4 and E = E = 28 ⋅ 2 − 14. At last, the results obtained 5 6 n 3 n 2 n M DS , x = 4 ⋅ 3 x + (4 ⋅ 3 − 4)x + (10 ⋅ 3 − 10), by means of the definitions of these degree-based 3 1 indices. ✷ Remark 7: By taking the special value of k in results n n n 4⋅3 10⋅3 −10 4⋅3 −4 PM DS = 5 4 6 , 1 1 of Theorem 7, we get M PD = 280 ⋅ 2 − 130, 1 2 40⋅3 −32 PM DS = 2 , 2 1 M PD = 296 ⋅ 2 − 140, 2 2 ReZG DS = 18 ⋅ 3 − 13, 1 1 46 77 ∗ n M PD = ⋅ 2 − , 2 2 3 12 278 46 ReZG DS = ⋅ 3 − , 2 1 15 3 n n � � 4 ⋅ 2 15 28 ⋅ 2 − 14 PD = + 16 ⋅ 2 − + , √ √ ReZG DS = 432 ⋅ 3 − 352. 3 5 3 1 182 W. GAO ET AL. Proof: By analysis of PAMAM dendrimer DS struc- Disclosure statement tures, we found that the edge set of DS can be divided e a Th uthors declare that there is no conflict of interests into three parts. regarding the publication of this paper. • E , d(u) = 1 and d(v) = 4; Funding • E , d(u) = d(v) = 2; • E (or E ), d(u) = 2 and d(v) = 4. This work was supported by the National Natural Science 6 8 Foundation of China [grant number 11401519]. Additionally, it is not hard to check that E = 4 ⋅ 3 , n n E = 10 ⋅ 3 − 10, and E = E = 4 ⋅ 3 − 4. Finally, 4 6 8 References the results deduced according to the definitions of these degree-based indices. ✷ Azari, M., & Iranmanesh, A. (2011). Generalized Zagreb index of graphs. Studia Universitatis Babes-Bolyai, 56, 59–70. Remark 8: Again, in view of taking the special value Bollobas, B., & Erdos, P. (1998). Graphs of extremal weights. of k in results of Theorem 8, we get Ars Combinatoria, 50, 225–233. Bondy, J. A., & Mutry, U. S. R. (2008). Graph theory. Berlin: M DS = 84 ⋅ 3 − 64, Spring. 1 1 Eliasi, M., & Iranmanesh, A. (2011). On ordinary generalized geometric-arithmetic index. Applied Mathematics Letters, n 24, 582–587. M DS = 88 ⋅ 3 − 72, 2 1 Eliasi, M., Iranmanesh, A., & Gutma, I. (2012). Multiplicative versions of first Zagreb index. 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Geology Ecology and Landscapes – Taylor & Francis
Published: Jul 3, 2017
Keywords: Theoretical chemistry; general sum connectivity index; general Randic index; H-Naphtalenic nanotube; PAMAM dendrimer
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