An Indian Journal FULL PAPER ABSTRACT KEYWORDS. Trade Science Inc. Hyper-wiener index of hexagonal mobius graph. [Type text] [Type text] [Type text]

Transcription

1 [Type text] [Type text] [Type text] ISSN : Volume 0 Issue BioTechnology 04 An Indian Journal FULL PAPER BTAIJ, 0(), 04 [ ] yper-wiener index of hexagonal mobius graph Jiayong Dou *, Yaya Wang, Wei Gao Department of Information Engineering, Binzhou Polytechnic, Binzhou, (CINA) School of Information Science and Technology, Yunnan Normal University, Kunming, (CINA) ABSTRACT yper-wiener index is an important topological index in theoretical chemistry. Physical chemical properties of material are closely related to this index. exagonal Mobius graphs are one type of molecular graphs embedded into the Mobius strip such that each face is a hexagon. In this paper, we obtain the yper-wiener index of the two classes of hexagonal Mobius graphs. KEYWORDS Chemical graph theory; Organic molecules; yper-wiener index; exagonal mobius graph; Automorphism. Trade Science Inc.

2 BTAIJ, 0() 04 Jiayong Dou et al. 63 INTRODUCTION The yper-wiener index, as an extension of Wiener index, is an important topological index in Chemistry. It is used for the structure of molecule. There is a very close relation between the physical, chemical characteristics of many compounds and the topological structure of that. The yper-wiener index is such a topological index and it has been widely used in Chemistry fields. The graphs considered in this paper are simple and connected. The vertex and edge sets of G are denoted by V(G) and E(G), respectively. The Wiener index is defined as the sum of distances between all unordered pair of vertices of a graph G, i.e., WG= ( ) d( u, v), { uv, } V ( G) where duv (, ) is the distance between u and v in G. The yper-wiener index WW is one of the recently distance-based graph invariants. That WW clearly encodes the compactness of a structure and the WW of G is define as: WW ( G) = ( duv (, ) + duv (, )). { u, v } V ( G ) { u, v } V ( G ) Some conclusions for yper-wiener index and Wiener index can refer to [-5]. Pan [6] deduced the formula of Wiener number and yper-wiener number of two types of polyomino systems. Tang [7] studied the Wiener indices of unicycles graphs. Firstly, it gave a formulation for calculating the Wiener index of an unicycles graphs according its struction. And then, in terms of this formulation, it characterized the graphs with the largest, the smallest, the second largest, the second smallest, the third largest and the third smallest Wiener indies among all the unicycles graphs. Xing et al., [8] determined the n-vertex unicyclic graphs of cycle length r with the smallest and the largest yper-wiener indices for 3 r n, and the n-vertex unicyclic graphs with the smallest, the second smallest, the largest and the second largest yper-wiener indices for n 5. Yuan [9] learned the special class of unicyclic graph. Feng et al., [0] determined the extremal bicyclic graphs with maximal and minimal hyper-wiener index. Chen [] investigated the properties of the Wiener index of unicyclic graphs, which are used to give a lower bound for the Wiener index of unicylic graphs of order β having perfect matching. Moreover, all extremal unicyclic graphs which attain the lower bound were characterized. Qi and Zhou [] determined the minimum yper-wiener index of unicylic graphs with given number of vertices and matching number, and characterized the extremal graphs. Feng and llic [3] presented sharp bounds for the Zagreb indices, arary index and hyper-wiener index of graphs with a given matching number, and we also completely determine the extremal graphs. Also, Du and Zhou [4] determined the minimum Wiener indices of trees and unicyclic graphs with given number of vertices and matching number respectively, and the extremal graphs are characterized. Let G=(V, E) and =(V, E ) be two graphs. G is a such graph, its vertex set is V V ', there exist an edge between (a, x) an (b, y) if and only if one of following condition holds: () ab E and x=y; () a=b and xy E. Let V(P )={0,,,-}. The hexagonal Mobius graph with length and width denoted by is defined as follows: deleting edges {(0, j+) (, j+) 0 j -} from,, P P then adding edges (,0)(0,-) and (0,0)(,-). The hexagonal Mobius graph with length + and width 3 denoted by is defined as follows: deleting edges {(0, j) (, j) 0 j } from, 3, + P3 P + then adding edges (0,0)(,), (,0)(,) and (,0)(0,).

3 63 yper-wiener index of hexagonal mobius graph BTAIJ, 0() 04 Figure :, and 3, + In this paper, we determine the yper-wiener index of above two hexagonal Mobius graphs. Main results and proof 8, = Theorem. WW ( =., ) 70, = (4i + ) + ( + ), 3 Proof. When = and =, and are isomorphism to following graphs (see Figure.), respectively. By the definition of yper-wiener index and directly computing, we have WW ( ) =8 and WW ( ) =70.,4,,4 When 3, the vertices of have two orbits under its automorphism group, and all the, vertices with degree in one orbit and all the vertices with degree 3 in the other orbit., Figure : The graphs isomorphism to and, respectively.,,4

4 BTAIJ, 0() 04 Jiayong Dou et al. 633 Taing any vertex v with degree, the sum of distance square from v to other vertex is calculated by (see left graph of Figure 3.) ( + + L + ) + 3 = 4 i + 5. Taing any vertex v with degree 3, the sum of distance square from v to other vertex is calculated by (see right graph of Figure 3.) L ( ) = 4 i. Figure 3 : Distance computing by taing any vertex with degree or 3 from, respectively. ence, we infer WW ( =, ) { { (4i + 5) + (4i )} + { [ ( + ) + ] + [ ( + ) ]}} = {{ (8 4)} {4 ( )}} i = (4 i + ) + ( + ).

5 634 yper-wiener index of hexagonal mobius graph BTAIJ, 0() 04 Figure 4 : The graph isomorphism to and the structure of, respectively. 3,3 3,5 0, = Theorem. WW ( =. 3, + ) 465, = + {8 i + ( )}, 3 4 Proof. When =, 3,3 is isomorphism to left graph of Figure 4. By the definition of yperwiener index and directly computing, we have WW ( =0. When =, the structure of is 3,3) 3,5 showed in right graph of Figure 4. We yield WW ( =465. 3,5) When 3, the vertices of have three orbits under its automorphism group, and all the 3, + vertices with degree and degree in boundary in the first and second orbit respectively, and all the inner vertices with degree 3 in the third orbit. 3, + Figure 5 : Distance computing by taing any vertex from. There are + vertices with degree 3 in boundary. Taing any vertex v, the sum of distance square from v to other vertex is calculated by (see middle graph of Figure 5.) { ( + + L+ ( + ) ) } + {( + + L+ ( + ) ) } ( + + ) + 4 = 6 i + 5. There are + vertices with degree in boundary. Taing any vertex v, the sum of distance square from v to other vertex is calculated by (see left graph of Figure 5.) +

6 BTAIJ, 0() 04 Jiayong Dou et al. 635 { ( + + L+ ( + ) ) } + {( + L+ ( + ) ) ( )} = 6 i There are + inner vertices with degree 3. Taing any vertex v, the sum of distance square from v to other vertex is calculated by (see right graph of Figure 5.) { ( + + L+ ( + ) ) } + {( + + L+ ) ( + )} + 3 = 6 i + 4( + ) + 6. ence, we infer WW ( = 3, + ) { [6i i i + 4( + ) + 6] + ( + )(9 + 6)} + = {8 i + ( )}. 4 ACKNOWLEDGEMENTS First, we than the reviewers for their constructive comments in improving the quality of this paper. This wor was supported in part by Binzhou Polytechnic Fund (0XYKT0) and (03XYKT08). We also would lie to than the anonymous referees for providing us with constructive comments and suggestions. REFERENCES [] Wei Gao, Li Shi; yper-wiener index of gear fan and gear wheel related graph, Manuscript. [] Wei Gao, Li Shi; Wiener index of gear fan graph and gear wheel graph, Asian Journal of Chemistry, 6(), (04). [3] Deqin Chen; The value of the Wiener index of the peptoids, J.Wuhan Inst.Chem.Tech., 7(), 8-84 (005). [4] Baohua Xing, Gaixiang Cai; The Wiener index of trees with prescribed diameter, Operations Research Transactions, 5(4), (0). [5] ua Wan, aizhen Ren; The Wiener index of a class of tricyclic graphs, Journal of Mathematical Study, 45(), 07- (0). [6] Yunjing Pan; Wiener number and hyper-wiener number of two types of polyomino systems, Journal of Mathematical Study, 46(3), (03). [7] Ziai Tang; The Wiener indices of unicyclic graphs, Master thesis of unan Normal University, (006). [8] Rundan Xing, Bo Zhou, Xuli Qi; yper-wiener index of unicyclic graphs, MATC Commun.Math.Comput.Chem., 66, (0). [9] ou Yuan; On the yper-wiener index of unicyclic graphs, Journal of Mathematical Study, 6(), 4-50 (03). [0] Lihua Feng, Weijun Liu, Kexiang Xu; The hyper-wiener index of bicyclic graphs, Manuscript. [] Yahong Chen; The Wiener Index of Unicyclic Graphs with Perfect Matching, Journal of Shanghai Jiaotong University, 44(6), (00). [] Xuli Qi, Bo Zhou; On the hyper-wiener index of unicyclic graphs with given matching number, Stud.Univ.Babes-Bolyai Math., 57(4), (0). [3] Lihua Feng, Alesandar Llic, Zagreb; arary and hyper-wiener indices of graphs with a given matching number, Applied Mathematics Letters, 3(8), (00). [4] Zhibin Du, Bo Zhou; Minimum Wiener indices of trees and unicyclic graphs of given matching number, MATC Commun.Math.Comput.Chem., 63, 0- (00). +