Wiener Index Of A Graph And Chemical Applications

Transcription

1 Internatonal Journal of hemtech Reearch ODE( USA): IJRGG ISS : Vol.5, o.4, pp , Aprl-June 013 Wener Index Of A Graph And hemcal Applcaton A.Vayabarath 1 *, G.S.G..Ananeyulu 1 Reearch Scholar, Appled Algebra Dvon, SAS, VIT Unverty, Vellore-14, Tamlnadu, Inda. Aocate profeor, Appled Algebra Dvon, SAS, VIT Unverty, Vellore-14, Tamlnadu, Inda. orre.author: vayabaratha010@vt.ac.n 1, ananeyulu.ggn@vt.ac.n Abtract: Wener ndex one of the topologcal ndce whch can be ued for analyzng ntrnc properte of molecule tructure n chemtry. In th paper we evaluate a wener ndex of a graph (molecular graph) n two way: one new method by ung uper edge-magc equence (SEMS) and the other a dfferent approach for extng method, ung Mnmal pannng tree at each vertex. By th, we can reduce laborou procedure n extng method by pannng tree algorthm, a many poble. Partcularly the tude on wener ndex of the molecular graph to analyze the tructure of organc molecule lke yclo alkane, Alkane-n-amne and Alkanen,n -damne through the SEMS have been preented n th paper. Key word: wener ndex, uper edge-magc equence, pannng tree. AMS ubect clafcaton: 051, 0578, Introducton: 1.1. Back ground of wener Index In chemtry the Wener ndex one of the mot thoroughly tuded, bet dtnguhed and mot frequently ued graph-theory-baed molecularhape decrptor [5 and 14]. Graph theory appled n the tudy of molecular tructure repreent an nterdcplnary cence, called chemcal graph theory or molecular topology. By ung tool taken from the graph theory, et theory and tattc t attempt to dentfy tructural feature nvolved n tructure-property actvty relatonhp [1 and 13]. The Wener ndex W the frt topologcal ndex to be ued n chemtry [14]. A topologcal ndce are number aocated wth chemcal tructure. There not a one-to-one connecton wth chemcal tructure and topologcal ndce, becaue everal graph may have the ame topologcal ndex. Varou topologcal ndce uually reflect molecular ze and hape. A topologcal repreentaton of a molecule called molecular graph. A molecular graph a collecton of pont repreentng the atom n the molecule and et of lne repreentng covalent bond. Thee pont are named vertce and lne are named edge n graph theory language. Frt mathematcal defnton of Wener ndex, baed on the concept of graph- theoretcal dtance a encoded n the dtance matrx [11] due to ooya [5] nce t ntaton the wener ndex wa ued n a numerou tructure-property tude [15].Wener ndex wa developed by the Amercan hemt arold Wener n 1947 and ued t to determne phycal properte of type of alkane known a paraffn [17 and 18]. Alkane are organc compound excluvely compoed of carbon and hydrogen.

2 A.Vayabarath et al /Int.J.hemTech Re.013,5(4) 1848 The name Wener number or Wener ndex nowaday n tandard ue n chemtry and ome tme encountered alo n the mathematcal lterature [9 and 10]. It mot tuded topologcal ndce both from a theoretcal pont of vew and applcaton.[1, 4 and 7] The ue of modern topologcal ndce n QSPR and QSAR begn wth the wener ndex..wener defned Wener ndex W (G) [3 and 8] a the um of mallet dtance between all vertce of the graph G W (G) = d( V,) V The Wener ndex W (G) of a graph G defned a the um of the half of the dtance between every par of vertce of G. 1 W ()() G n n 1 1 d v v 1.. Outlne of the paper The ret of the paper organzed a follow. In Secton, we compute wener ndex of a graph by the followng way: () Through Super edge -magc equence and ()General method-mnmal Spannng Tree Technque at each vertex. Th a new approach for extng one. It tme conumng method for fndng wener ndex of a graph havng huge number of vertce, becaue there are o many pannng tree algorthm avalable. More than that, method () applcable only for thoe graph (molecular graph) havng SEMS. All the molecular graph can t have SEMS, but t mut have wener ndex. In that ntance we can apply the method (). In Secton 3 deal wth dentfcaton of the chemcal compound lke yclo alkane, Alkane-n-amne and Alkane-n, n - damne through the SEMS n addton of tude on wener ndex. Th the ntal approach to degn equence for chemcal compound by ung graph theory. It could be ued a a ecurty code n hemcal Reearch. Fnally, concludng remark are made n Secton 4.. Super edge-magc equence and Wener Index Varou author have ntroduced labelng (valuaton) concept. Kotzg and Roa ntroduced the concept of magc valuaton [6]. Rngel and Llado [14] called th type of valuaton a edge magc labelng. Enomoto et. al. [] retrcted the noton of edge magc labelng of a graph to obtan the defnton of uper edge magc labelng. We ntroduced the concept and properte of uper edge-magc equence n [16]. In the ame paper we dcued the followng theorem: Theorem:.3.1 A graph a uper edge-magc graph f and only f t ha uper edge- magc equence..1. ontructon of SEMS from a fnte et of natural number In the current paper to reach frutful reult n all apect, we convert any equence of natural number nto SEMS a follow. So that molecule tructure can be extracted ether from natural number equence or drectly from any SEMS f poble. onveron of any equence of natural number to uper edge-magc equence acheve by purung the followng tep. Step1: Let (x 1, x,, x n ) be any equence of natural number whoe length ay q. Step: Fnd (y 1, y,, y n ) be another equence of number wth ame length uch that x + y conecutve wth x 1 +y 1 x n +y n, for 1 n. Step3: Set L = mn {x, y 1 n} and U = max { x, y 1 n } ow (L 1, L,, L n ) denote the lower end vertce for each ee(g) and (U 1,U,,U n ) denote the upper end vertce for each ee(g). Then by defnton of SEMS (L 1, L,, L n ) repreent a uper edge-magc equence and the correpondng graph repreentaton (L 1, L,,L n ). (U 1, U U n ) Example.1.1: Suppoe (3,4,5,6,1) be gven any equence of natural number. Accordng the above tep, (x 1, x,, x 5 )= (3,4,5,6,1) of length q=5. (y 1, y,, y 5 ) = (7,5,3,1,5) wth (L 1, L,, L 5 ) = (3,4,3,1,1) and (U 1,U,,U n ) = (7,5,5,6,5)

3 A.Vayabarath et al /Int.J.hemTech Re.013,5(4) 1849 (3,4,3,1,1) (7,5,5,6,5) th wll produce uper edge-magc graph a hown n fg.1 and the correpondng SEMS (3,4,3,1,1) 4 (v). In any column (row), If all the entre are zero then the correpondng vertex a olated vertex. Then we ay that the correpondng uper edgemagc graph ha mnmum defcency [16]. (v). In any column (row) ha at leat one zero then the graph not connected. We ay that t ha more than one component. Fg.1.. alculaton of Wener ndex ung SEMS Ether from the graph (or) equence, we wll get one SEMS. To proceed for further to defne the followng Bond Matrx. Defnton:..1. (Bond matrx) Bond matrx a matrx whch gve relaton between every par among the element of the et {1,, 3,, p}. That, t an upper trangular(lower trangular) matrx of uper edge-magc equence n whch um of all the entre equal to wener ndex of the correpondng graph and defned a follow: W(G) =w, n-number of vertce of a graph n n Where w can be defned for < and w = whenv and v arenot related n permutatonmatrx k when mn. noof relatonnbetweenv v nthe permutatonmatrx Where permutaton matrx the equence repreentaton of a graph mentoned n ecton.1. Obervaton of Bond matrx..: Formaton of a bond matrx produce complete nformaton about a graph wthout drawng. Bond matrx a ymmetrc matrx. (). umber of one n a row (column) of a wener matrx equal to degree of a vertex. (). Total number of one n an upper trangular matrx equal to number of edge of a graph. (). Order of the matrx equal to number of vertce n a graph..e., order of the matrx= n= max {U / 1 n} oncrete Example..3: onder the uper edge-magc equence a n the example Sum of all entre of upper trangular matrx (lower trangular)= =5 Ung Bond matrx, Wener ndex of fg alculaton of Wener ndex by Mnmal pannng tree In general, all graph (molecular graph) need not have uper edge-magc equence, o wener ndex for thoe graph are calculated by extng method. But extng method tedou for thoe graph havng large number of vertce. So we would lke to ugget a new technque baed on pannng tree, becaue many pannng tree algorthm are avalable. Th can be performed by Mnmal pannng tree technque at each vertex and a follow. Procedure: In the defnton of Wener ndex, W (G) = d( V,) V. At v 1, dentfy a pannng tree from G whch havng hortet path from other vertce to v 1, from th d( V1,) V can be calculated and 1 dented by S 1 for >1. Smlarly we dentfy pannng tree at other vertce of G ndependently; from them the other component of W (G) can be calculated.

4 A.Vayabarath et al /Int.J.hemTech Re.013,5(4) 1850 ow we fnd dtance matrx of G ung above mnmal pannng tree then for for for for W (G) = oncrete Example.3.1: hooe a graph G whch han t SEMS. For th, conder a cycle on 6 vertce whch doen t have uper edge-magc labelng. That graph nothng but exagon and t mnmal pannng tree wth repect to t vertce are: v 6 v 6 v 1 v Mt(v v ) 1 v v5 v 1 v G Mt( ) v 6 v 3 v 3 v 6 In the above graph doted edge for <, d (v, v ) are not condered. Snce they are calculated n the prevou pannng tree. The vertce whch are marked by whte crcle ndcate that calculaton of mnmal pannng tree wth repect to that vertex. Wener ndex of exagon a follow: v 1 v v 1 v 1 v v5 Mt(v 1 ) Mt(v 3 ) v Mt( ) v 6 v 3 v 3 v 6 v for < and 1 6; 1 5 Sum=7 Wener ndex of exagon = Identfyng a chemcal compound through the uper edge-magc equence: Each SEMS gve one uper edge-magc graph, but all uper edge-magc graph need not repreent a chemcal compound. So that t poble to dentfy ome chemcal compound through the uper edgemagc equence. In a graph theory language, the degree of each vertex repreentng the valency (number of bond ncdent on a vertex) of the molecule n molecular graph. umber of covalent Bond: 4 for carbon; 3 for ntrogen; for oxygen and 1 for hydrogen Rule for Identfcaton Suppoe (a 1, a, a 3,.a, a q-, a q-1, a q ) repreent SEMS for a chemcal compound and th equence ha two part eparated by a broken lne. otaton and repreentaton of equence are followed n paper[16] Frt part of the equence alway generate a pendent edge n whch vertce are atom of hydrogen. Second part of the equence generate ether a cycle or a chan of a chemcal compound dependng upon the SEMS. Maxmum number appeared n the equence denoted by total number of atom and length of the equence denoted by total number of bond of the compound. Wener ndex of a chemcal compound depend upon the econd part of the SEMS and t can be calculated by ung Bond matrx. Snce wener ndex are calculated only for ydrogen uppreed graph. The compound name correpondng equence wth ther property are dcued n Table:3.1.1.

5 A.Vayabarath et al /Int.J.hemTech Re.013,5(4) Reult and Dcuon ompound- 1: yclo Pentane Sequence Repreentaton: (5,, 4, 1, 3, 5,, 4, 1,3, 3,,,1,1) Maxmum number appeared n the above equence 15 =Total number of Atom and Length of the equence 15 =Total number of Bond of yclo Pentane. Structure: Wener Index: Ung Bond matrx Wener ndex of yclo Pentane 15. ompound-: Butane-1-amne Sequence Repreentaton: (3, 5,, 4, 1, 3, 5,, 4, 1, 3, 3,,, 1) Maxmum number appeared n the above equence 16 =Total number of Atom and Length of the equence 15 =Total number of Bond of Butane-1-amne. Structure: Wener Index: Ung Bond matrx Wener ndex of Butane-1-amne 0. ompound-3: Propane -1, 3- Damne Sequence Repreentaton: (5,, 4, 1, 3, 5,, 4, 1, 3, 3,,, 1) Maxmum number appeared n the above equence 15 =Total number of Atom and Length of the equence 14 =Total number of Bond. Structure: Wener Index: Wener ndex of Propane 1,3- Damne 0. The detal of reult and dcuon of a chemcal compound are preented n Table : oncluon Th paper contaned two mportant mechanm: one calculaton of wener ndex that nclude two way: frt the new method ung SEMS and the other method a new technque for extng one. The other part of the mechanm the dentfcaton of the chemcal compound through the SEMS. Th approach wll help to tranmt the chemcal formula nto equence.

6 A.Vayabarath et al /Int.J.hemTech Re.013,5(4) 185 Table:3.1.1 S.o. ompound ame SEMS for a ompound Property of SEMS n[16] Partcular cae for equence when =. 1. yclo Alkane ((+1,,,-1,-1,.,,1,+1) (),+1,,,-1, -1,,1,1) α*=+, q=3(+1) p=3(+1) p = q (uncycle) (5,, 4, 1, 3, 5,, 4, 1,3, 3,,, 1,1) yclo Pentane. Alkane-n-Amne (+1,(+1,,,-1, -1,,1,+1) (),+1,,, -1,-1,, 1) 3. Alkane- n, n -Damne ((+1,,,-1, -1,,1,+1) (),+1,,, -1,-1,, 1) α*=+3 q=3(+1) p=3(+1)+1. p = q+1(tree) α*=+3 q=3(+1)-1 p=3(+1). p = q+1(tree) ( 3, 5,, 4, 1, 3, 5,, 4, 1, 3, 3,,, 1) - Butane-1-Amne (5,, 4, 1, 3, 5,, 4, 1, 3, 3,,, 1) - Propane-1, 3- Damne Table : S. Reult for chemcal o. compound General cae ompound - 1 ompound - ompound - 3 Specfc cae for = General cae Specfc cae for = General cae Specfc cae for = 1. umber of Atom 3(+1) 15 3(+1) (+1) 15. umber of Bond 3(+1) 15 3(+1) 15 3(+1) yclomatc number umber of atom n the cycle/ chan umber of bond n the cycle /chan hemcal formula for +1 ( a compound +1) 7. IUPA ame yclo Alkane 5 10 yclo Pentane (()+3) Alkane-namne - ) ( ) -1-3 Butane-1- Alkane-n,n - Propane-1, 3- amne damne Damne 8. Wener ndex calculated by bond matrx ung the cycle calculated by bond matrx ung the chan 0 calculated by bond matrx ung the chan 0

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