FOUR VERTEX-DEGREE-BASED TOPOLOGICAL INDICES OF CERTAIN NANOTUBES

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1 U.P.B. Sci. Bull., Series B, Vol. 80, Iss.,, 08 ISSN FOUR VERTEX-DEGREE-BASED TOPOLOGICAL INDICES OF CERTAIN NANOTUBES Xu LI, Jia-Bao LIU *, Muhammad K. JAMIL 3, Aisha JAVED 4, Mohammad R. FARAHANI 5 Let G be a simle connected grah with vertex set V(G) and edge set E(G). The number of elements in V (G) is called the order of G, denoted as V (G), and the number of elements in E(G) is called the size of G, denoted as E(G). A toological index is a numerical descritor of a molecular structure derived from the corresonding molecular grah. Recently, four vertex-degree based grah invariants, recirocal Randić (RR), reduced recirocal Randić (RRR), reduced second Zagreb (RM) and forgotten (F) indices was defined by I. Gutman. In this article, we comute these indices of HC5C7[;q] and SC5C7[;q] Nanotubes. Keywords: Recirocal Randić index, Reduced recirocal Randić index, Reduced second Zagreb index, Forgotten index, HC5C7[;q] nanotube, SC5C7[;q] nanotube. Introduction Let G(V(G),E(G)) be a simle connected grah, where V(G) and E(G) reresents the vertex set and edge set, resectively. Number of elements in V(G) is called the order of G, denoted as V(G), and the number of elements in E(G) is called the size of G, denoted as E(G). For a vertex u V ( G), the degree of u in G, d(u), is the number of vertices adjacent with u in grah G. A toological index is a numerical descritor of the molecular structure derived from the corresonding molecular grah. Many toological indices are widely used for quantitative structure-roerty relationshi (QSPR) and quantitative structure-activity relationshi (QSAR) studies. The concet of Deartment of Mathematics and Physics, Hefei University, Hefei, China, xuli@hfuu.edu.cn School of Mathematics and hysics, Anhui Jianzhu University, Hefei, China, liujiabaoad@63.com 3 Deartment of Mathematics, Rihah Institute of Comuting and Alied Sciences (RICAS), Rihah International University, Lahore, Pakistan, m.kamran.sms@gmail.com 4 Abdus Salam school of Mathematical Sciences, GC University, Lahore, Pakistan, aaishajaved@gmail.com 5 Deartment of Alied Mathematics, Iran University of Science and Technology, Narmak, Tehran, Iran, MrFarahani88@Gmail.com, Mr_Farahani@Mathde.iust.ac.ir

2 78 Xu Li, Jia-Bao Liu, Muhammad K. Jamil, Aisha Javed, Mohammad R. Farahani toological index came from work done by Harold Wiener in 947 while he was working on boiling oint of araffin. In chemical grah theory, several vertex-degree based toological indices have been and are currently considered and alied in QSPR=QSAR studies. Among them first Zagreb index M, the second Zagreb index M are the oldest and most thoroughly investigated. Following are there definitions: M ( G) d( u) uv ( G) M ( G) d( u) d( v) uve ( G) In addition, we mention some other vertex-degree based toological indices widely used in chemical literature. Randić index was roosed by the chemist Milan Randić [] in 975. For a grah G it is defined as, RG ( ) d( u) d( v) uve ( G) In 998 Bollobas and Erdos [] relaced by any real number \alha to generalize this index, which is known as the general Randić index. R ( G) ( d( u) d( v)) uve ( G) The zero th -order general Randić index, 0 R ( G) was defined in[3] as 0 R ( G) d( u) uv ( G) Zhou et. al. in [4, 5] roosed the sum-connectivity index. The sum-connectivity index is obtained from Randić index by relacing d(u)d(v) by d(u)+d(v), ( G ) uv E ( G ) d ( u) d ( v ) The concet of sum-connectivity index was extend to the general sumconnectivity index in ( G) ( d( u) d( v)) uve ( G) In 980, Fajtlowicz defined an invariant of the Randić index called the harmonic index, defined as HG ( ) uve ( G) d( u) d( v) For recent results on vertex-degree based toological indices, readers can see the aer series [6-8].

3 Four vertex-degree-based toological indices of certain nanotubes 79 Recently, Gutman et.al. resented four vertex-degree based grah invariants, that earlier have been considered in the chemical and/or mathematical literature, but, that evaded the attention of most mathematical chemists. The new/old toological indices studied by I. Gutman et. al. are the following [9, 0]: The recirocal Randić index is defined as RR( G) d( u) d( v) uve ( G) The reduced recirocal Randić index is defined as RRR ( G ) ( d ( u ) )( d ( v ) ) uv E ( G ) The reduced second Zagreb index is defined as RM ( G ) ( d ( u) )( d ( v ) ) uv E ( G ) In a study on the structure-deendency of the total -electron energy, beside the first Zagreb index, it was indicated that another term on which this energy deends is of the form F G d v d u d v 3 ( ) ( ) ( ( ) ( ) ) v V ( G ) uv E ( G ) Recently, this sum was named forgotten index, or shortly the F index. Carbon nanotubes are tyes of nanostructure which are allotroes of carbon and having a cylindrical shae. Carbon nanotubes or molecular grahs SC5C7[,q] and HC5C7[,q] are two family of nanotubes, such that their structure are consist of cycles with length five and seven by different comound. In other words, a C5C7 net is a trivalent decoration made by alternating C5 and C7. It can cover either a cylinder or a torus. For a review, historical details and further bibliograhy see the 3-dimensional lattice of VC5C7[,q] and HC5C7[,q] nanotubes in Fig. and their -dimensional lattice in Fig. and Fig. 4, resectively. For further study reader can see [-36]. Fig.. The Molecular grah of SC 5C 7 (A) and HC 5C 7 (B) nanotubes

4 80 Xu Li, Jia-Bao Liu, Muhammad K. Jamil, Aisha Javed, Mohammad R. Farahani. Main Results In this section, we focus on the structures of two families of carbon nanotubes SC5C7[,q] and HC5C7[,q] are of nanotubes and we will comute the recirocal Randić, reduced recirocal Randić, reduced second Zagreb and forgotten indices of HC5C7 and SC5C7 nanotubes. To comute certain toological indices of thee nanotubes, we will artition the edge set based on degrees of end vertices of each edge of the grah... Nanotube HC5C7[,q],(,q>) In this section, we calculate some toological indices of HC5C7[;q] nanotube. The two dimensional lattices of HC5C7[;q], in which is the number of hetagons in first row and q is the number of eriods in whole lattice. A eriod consist of four rows, a m th eriod is shown in Fig.. This nanotube is a C5C7 net and its -dimensional lattice is constructed by alternating C5 and C7 following the trivalent decoration as shown in Fig. and Fig. 3. This tye of attern of C5 and C7 can either cover a cylinder or a torus. There are 6 vertices in one eriod of the lattice and vertices are joined to the end of the grah, so V(HC5C7[;q]) =6q+. Similarly, there are 4 edges in one eriod and extra edges which are joined to the end of the grah in these nanotubes, so we have E(HC5C7[;q]) =4q-. Theorem. Consider the grah HC5C7[;q] nanotube, then RR ( HC C [, q ]) (36q 4 6 5) 5 7 RRR ( HC C [, q ]) (q 5)4 5 7 RM ( HC C [, q]) (4q )4 5 7 F ( HC 5C 7[, q]) (08 q 9)4 (8) Proof. Consider the nanotube G=HC5C7[;q], where is the number of hetagons in one row and q is the number of eriods in whole lattice. To obtain the final results we artition the edge set based on degrees of end vertices of G. There are two artitions of edge set corresond to their degrees of end vertices which are E { uv E ( G ) d ( u ) and d ( v ) 3} E { uv E ( G ) d ( u ) 3 and d ( v ) 3} (5) (6) (7)

5 Four vertex-degree-based toological indices of certain nanotubes 8 Fig. : The grah of HC 5C 7[;q] nanotube with =3 and q=3 Fig. 3: The m th eriod of HC 5C 7 nanotube The number of edges in E and E are 8 and 4q-0. Now, we can comute the RR; RRR; RM and F to comute these indices for G. Since, RR ( G ) d ( u ) d ( v ) d ( u ) d ( v ) d ( u ) d ( v ) uv E ( G ) uv E( G ) uv E ( G ) 8 3 (4q 0 ) 3 3 (36q 4 6 5) which is the required (5) result. RRR ( G ) ( d ( u ) )( d ( v ) ) uv E ( G ) ( d ( u ) )( d ( v ) ) ( d ( u ) )( d ( v ) ) uv E( G ) uv E ( G ) 8 ( )(3 ) (4q 0 ) (3 )(3 ) (q 5)4 which is the required (6) result.

6 8 Xu Li, Jia-Bao Liu, Muhammad K. Jamil, Aisha Javed, Mohammad R. Farahani RM ( G ) ( d ( u ) )( d ( v ) ) uv E ( G ) ( d ( u ) )( d ( v ) ) ( d ( u ) )( d ( v ) ) uv E( G ) uv E ( G ) 8 ( )(3 ) (4q 0 )(3 )(3 ) (4q )4 which is the required (7) result. F ( G ) ( d ( u ) d ( v ) ) uv E ( G ) ( d ( u ) d ( v ) ) ( d ( u ) d ( v ) ) uv E( G ) uv E ( G ) 8 ( 3 ) (4q 0 )(3 3 ) (08q 9)4 which is the required (8) result, and the roof is comlete... Nanotube SC5C7[,q],(,q>) In this section, we comute the certain vertex-degree based toological indices of SC5C7[;q] nanotube. This nanotube is also C5C7 net is constructed by alternating C5 and C7 following the trivalent decoration as shown in Fig. 4 and Fig 5. This tye of tiling can either cover a torus or cylinder too. The SC5C7[;q], in which is the number of hetagons in each row and q is the number of eriods in whole lattice. A eriod consists of three rows as in Fig. 5 in which m th eriod is shown. There are 8 vertices in one eriod of the lattice so V(SC5C7[;q]) =8q. Similarly, there are edges in one eriod and extra edges which are joined to the end of the grah so E(SC5C7[;q]) =q. Theorem 3. Consider the grah of SC5C7[;q] nanotube, then F ( SC C [, q]) (54q 9)4 5 7 RM ( SC C [, q]) (48q 3) 5 7 RRR ( SC C [, q ]) (4q 6 7) 5 7 RR ( SC C [, q ]) (36q 6 6 5) 5 7 Proof. Consider the nanotube G=SC5C7[;q], where is the number of hetagons in one row and q is the number of eriods in whole lattice. To obtain the final results we artition the edge set based on degrees of end vertices of G. There are three artitions of edge set corresond to their degrees of end vertices which are (9) (0) () ()

7 Four vertex-degree-based toological indices of certain nanotubes 83 E { uv E ( G ) d ( u ) and d ( v ) } E { uv E ( G ) d ( u ) and d ( v ) 3} E { uv E ( G ) d ( u ) 3 and d ( v ) 3} 3 Fig. 4: The grah of SC 5C 7[;q] nanotube with =4 and q=4. Fig. 5: The m th eriod of SC 5C 7 nanotube. The number of edges in E; E and E3 are ; 6 and q-9. Now, we are able to aly the formula of RR; RRR; RM and F to comute these indices for G. Since, RR ( G ) d ( u ) d ( v ) uv E ( G ) d ( u ) d ( v ) d ( u ) d ( v ) d ( u ) d ( v ) uv E ( G ) uv E ( G ) uv E 3 ( G ) 6 3 (q 9 ) 3 3 (36q 6 6 5) which is the required (9) result.

8 84 Xu Li, Jia-Bao Liu, Muhammad K. Jamil, Aisha Javed, Mohammad R. Farahani RRR ( G ) ( d ( u ) )( d ( v ) ) uv E ( G ) ( d ( u ) )( d ( v ) ) ( d ( u ) )( d ( v ) ) ( d ( u ) )( d ( v ) ) uv E ( G ) uv E ( G ) uv E 3 ( G ) ( )( ) 6 ( )(3 ) ( q 9 ) (3 )(3 ) (4q 6 7) which is the required (0) result. RM ( G ) ( d ( u ) )( d ( v ) ) uv E ( G ) ( d ( u ) )( d ( v ) ) ( d ( u ) )( d ( v ) ) ( d ( u ) )( d ( v ) ) uv E ( G ) uv E ( G ) uv E 3 ( G ) ( )( ) 6 ( )(3 ) ( q 9 )(3 )(3 ) (48q 3) which is the required () result. F G d u d v ( ) ( ( ) ( ) ) uv E ( G ) ( d ( u ) d ( v ) ) ( d ( u ) d ( v ) ) ( d ( u ) d ( v ) ) uv E ( G ) uv E ( G ) uv E 3 ( G ) q ( ) 6 ( 3 ) ( 9 )(3 3 ) (54q 9)4 which is the required () result, and the roof is comlete. 3. Oen Problems and Concluding Remarks Recently, Gutman et. al. (re)introduced some new/old degree based toological indices and show that these toological indices have their own significance in the study of chemical grah theory. These indices are recirocal Randić index, reduced recirocal Randić index, reduced second Zagreb index and forgotten index. In this aer, we found these new/old toological indices of some nano-structure. These indices have not much studied u to now, so one can find the results for rest of nano-structures e.g. HAC5C7, HAC5C6C7, and Dendrimers. R E F E R E N C E S. M. Randić, On characterization of molecular branching, J. Am. Chem. Soc., 97 (975) B. Bollobas, P. Erdos, Grahs of extremal weights, Ars Combin., 50 (998) H. Hua, H. Deng, Unicycle grahs with maximum and minimum zero th -order general Randić indices, J. Math. Chem., 4 (007) 73-8.

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