The Degree Distance of Armchair Polyhex Nanotubes

Transcription

1 Applied Mathematical Sciences, Vol. 5, 20, no. 6, The Degree Distance of Armchair Polyhe Nanotubes Shubo Chen, Jianguang Yang, Fangli Xia and Qiao Chen Department of Mathematics, Hunan City University, Yiyang, Hunan 43000, P.R. China Abstract The degree distance of a graph which is a degree analogue of the Wiener inde of the graph. Let G = TUVC 6 [2p, q] be the carbon nanotubes covered by C 6, formulas for calculating the degree distance of armchair polyhe nanotubes TUVC 6 [2p, q] are provided. Mathematics Subject Classification: 05C05, 05C2 Keywords: Distance, degree distance, armchair polyhe nanotube Introduction Let G =(V,E) be a simple connected graph with the verte set V and the edge set E, d() denotes the degree of the verte. For any, y V, d(, y) denotes the distance (i.e.,the number of edges on the shortest path) between and y and we define D() = d(, y). The Wiener inde of the graph G, y V (G) is equals to the sum of distances between all pairs of vertices of the respective molecular graph, i.e., W (G) = d(, y) () {,y} V (G) The degree distance of a graph which is a degree analogue of the Wiener inde of the graph, was introduced by Dobrynin and Kochetova [2] and Gutman [3] as a weighted version of the Wiener inde. It was defined as DD(G) = V (G) d()d() = {,y} V (G) d(, y)(d()+d(y)) (2) This parameter has been investigated by several reports. Actually, when G is a tree on n vertices, the parameter is closely related as DD(G) =

2 262 S. Chen, J. Yang, F. Xia, Q. Chen 2W (G) n(n ), for this review see [3]. In [4], A. I. Tomescu characterized the connected unicyclic and bicyclic graphs in terms of the degree sequence, as well as the graphs in these classes minimal with respect to the degree distance are given. In [5], O. Bucicovschi and S. M. Cioabă studied the degree distance of graphs with given order and size, and determined the minimum degree distance of a connected graph of order n and size e. More results in this direction can be found in Refs. [6-0]. Carbon nanotubes were discovered in 992 by Iijima [] as multi-walled structures. Two years later, two groups independently discovered the singlewalled carbon nanotubes [2,3]. In 996, Smalley s group synthesized the aligned single-wall nanotubes [4]. As pointed out by Smalley, a carbon nanotube is a carbon molecule with almost alien property of electrical conductivity and super-steel strength. It is epected that carbon nanotubes can be widely used in many fields. Nanotubes and fullerenes are promising candidates in the development of nanodevices and super-strong composites. 2 Preliminary Notes Let G = TUVC 6 [2p, q] denotes an arbitrary armchair polyhe nanotube in terms of the circumference p and the length q. Fork( k q) is the various levels. An armchair polyhe nanotube with the parameters p = 0 is shown in Figure. Figure : The nanotube TUVC 6 [20,q] Figure 2. An armchair polyhe lattice with p = 4 and q = 7

3 Degree distance of armchair polyhe nanotubes 263 Note that the degrees of vertices in an arbitrary armchair polyhe nanotube are 2 or 3, then the Wiener inde W (G) can be written as W (G) = d G (u, v)+ d G (u, v)+ d(u),d(v)=3 The degree distance of G = TUVC 6 [2p, q] can be written as DD(G) =2 d G (u, v)+ 5 2 d G (u, v)+3 d(u),d(v)=3 d G (u, v) (3) d G (u, v) (4) Further, from the equations (3) and (4), the degree distance of an armchair nanotube can be rewritten as DD(G) =3W (G) Let W 2 (G) = d G (u, v) 2 d G (u, v), W 2,3 (G) = d G (u, v) (5) d G (u, v). Therefore, in our calculation we have to calculate W 2 (G), W 2,3 (G), since W (G) can be obtained from [7]. Now, we derive a formula for calculating the degree distance of TUVC 6 [2p, q]. 3 Main Results Lemma 3..LetG = TUVC 6 [2p, q], the Wiener inde of G is 2 p(24p2 q 2 +2q 4 8q 2 ), for q p and q is even; 2 p(24p2 q 2 +2q 4 8q 2 +6), for q p and p is even, q is odd; 2 p(24p2 q 2 +2q 4 8q 2 6), for q p, p, q are odd; W (G) = 2 p[p2 (2q 2 2p 2 +8)+8pq(p 2 + q 2 2)], for q p and p is even; 2 p[p2 (2q 2 2p 2 +8)+8pq(p 2 + q 2 2) 6], for q p and p is odd. Let s 0k be the sum of distances from v to all other vertices at level k, where v is a verte of degree 2 at level, α = +( )p, β = +( )p. Then 2 2 2p 2 + α k =; 2p s 0k = 2 β +(k ) 2, k p and k is even; 2p 2 + α +(k ) 2, k p and k is odd p 2 +2p(k ), k p. Let γ = ( ) q.

4 264 S. Chen, J. Yang, F. Xia, Q. Chen Theorem 3.2 Let G = TUVC 6 [2p, q], () p is even, 6 p{72p2 q 2 +6q 4 48p 2 q 8q 3 2q 2 +8q +3γ}, p>q; DD(G) = 3 p2( 3p 3 2p 2 q 8pq 2 2q 3 +4p 2 +2q 2 +2pq 6p +2q 4 ),p q. (2) p is odd, 6 p{72p2 q 2 +6q 4 48p 2 q 8q 3 2q 2 +8q 3γ}, p>q; DD(G) = 3 p( 3p 4 2p 3 q 8p 2 q 2 2pq 3 +4p 3 +2q 2 +2pq 2 +2p 2 q 6p 2 +2pq 4p +3 ), p q. Proof. We consider W 2 (G) and W 2,3 (G) and we divide them into two cases. Case. The number of W 2 (G). Subcase.. p q. Subcase... u and v are lying at level and level q. Subcase... when p is odd. W 2 (G) = d( i, j )+ d( qi, qj )=2p(2p 2 ) i<j 2p i<j 2p Subcase...2. when p is even. W 2 (G) = d( i, j )+ d( qi, qj )=2p 2p 2 i<j 2p i<j 2p Subcase..2. u is lying at level, v is lying at level q; u is lying at level q, v is lying at level. W 22 (G) = 2p 2p d( i, qj )=2p(p 2 +2pq 2p) i= j= Summing up, we arrive at, { 6p W 2 (G) =W 2 (G)+W 22 (G) = 3 +4p 2 (q ) 2p p is odd; 6p 3 +4p 2 (q ), p is even. Subcase.2. p q. Similar to the discussion in Subcase., we have, W 2 (G) = 8p 3 +2p(q ) 2 4p 8p 3 +2p(q ) 2 2p 8p 3 +2p(q ) 2 2p, 8p 3 +2p(q ) 2, p,q are odd; p,q are even; p is odd, q is even; p is odd, q is even. Case 2. The number of W 2,3 (G). Subcase 2.. p q. Subcase 2... when p is even.

5 Degree distance of armchair polyhe nanotubes 265 W 23 (G) = 2 2p { p (k ) 2 +(2p 2 ) p 2 +2p2 ( p ) 2 + q [p 2 +2p(k )] } k=p+ = 4 3 p4 +4p 3 q +4p 2 q 2 0p 3 2p 2 q p2 Subcase when p is odd. W 23 (G) = 2 2p { p (k ) 2 +(2p 2 +2p 2 ) p [p q k=p+ +2p(k )] } = 4 3 p4 +4p 3 q +4p 2 q 2 0p 3 2p 2 q p2 +2p Subcase 2.2. p q. Subcase when p is even. Subcase when q is even. W 23 (G) = 2 2p { p (k ) 2 +(2p 2 ) ( q ) + 2 2p2 ( q 2 )} = 8p 3 q pq3 6p 3 6pq pq 3 Subcase when q is odd. W 23 (G) = 2 2p { p (k ) 2 +(2p 2 ) ( q 2 )+2p2 ( q ) } 2 = 8p 3 q pq3 6p 3 6pq pq 2p 3 Subcase when p is odd. Subcase when q is even. W 23 (G) = 2 2p { p (k ) 2 +(2p 2 ) ( q ) + 2 2p2 ( q 2 )} = 8p 3 q pq3 6p 3 6pq pq 3 Subcase when q is odd. W 23 (G) = 2 2p { p (k ) 2 +2p 2 q +(2p 2 ) ( q ) } 2 2 = 8p 3 q pq3 6p 3 6pq pq 2p 3 Combining the above cases, we obtain the results. ACKNOWLEDGEMENTS. Project supported by the Research Foundation of Education Bureau of Hunan Province of China(Grant No: 0B05) and the Science and Technology of Hunan City University(Grant No. 200j006). References [] H. Wiener, Structural determination of paraffin boiling points, J. Am. Chem. Soc. 69(947), [2] A. A. Dobrynin, A. A. Kochetova, Degree distance of a graph: a degree analogue of the Wiener inde, J. Chem. Inform. Comput. Sci. 34 (994),

6 266 S. Chen, J. Yang, F. Xia, Q. Chen [3] I. Gutman, Selected properties of the Schultz molecular topological inde, J. Chem. Inform. Comput. Sci. 34 (994), [4] A. I. Tomescu, Unicyclic and bicyclic graphs having minimum degree distance, Discrete Appl. Math. 56 (2008), [5] O. Bucicovschi, S. M. cioabǎ, The minimum degree distance of graphs of given order and size, Discrete Appl. Math, in press [6] I. Tomescu, Properties of connected graphs having minimum degree distance, Discrete Mathematics. 309(2009), [7] M. Schocker, On degree sequences of graphs with given cyclomatic number, Publ. Inst. Math. (N.S.) 69 (200) [8] P. Dankelmann, I. Gutman, S. Mukwembi, H. C. Swart, On the degree distance of a graph, Discrete Appl. Math. 57 (2009), [9] S. Chen, Z. Guo, A Lower Bound on the Degree Distance in a Tree, Int. J. Contemp. Math. Sciences, 5(200), [0] I. Tomescu, Some etremal properties of the degree distance of a graph, Discrete Appl. Math. 98 (999), [] S. Iijima, Helical microtubles of graphitic carbon, Nature (London), 354(99), [2] S. Iijima and T. Ichlhashi, Single-shell carbon nanotube of -nm diameter, Nature (London), 363(993), [3] D. S. Bethune, C. H. Kiang, M. S. Deries, G. Gorman, R. Savoy, J. Vazquez, and R. Beters, Cobalt-catalysed growth of carbon nanotubes with single-atomic-layer walls, Nature (London), 363(993), [4] A. Thess, et al., Crystalline ropes of metallic carbon nanotubes, Science, 273(996), Received: August, 200