Estrada Index of Benzenoid Hydrocarbons

Transcription

1 Estrada Idex of Bezeoid Hydrocarbos Iva Gutma ad Slavko Radeković Faculty of Sciece Uiversity of Kragujevac P. O. Box Kragujevac Serbia Reprit requests to Prof. I. G.; Fax: ; Z. Naturforsch. 62a (2007); received December A structure-descriptor EE recetly proposed by Estrada is examied. If λ 1 λ 2...λ are the eigevalues of the molecular graph the EE = e λ i. I the case of bezeoid hydrocarbos with carbo atoms ad m carbo-carbo ( bods EE is foud to be accurately approximated by meas of / ) the formula a 1 cosh +a 2 wherea ad a 2 = 0.64 are empirically determied fittig costats. Withi classes of bezeoid isomers (which all have equal ad m) the Estrada idex is liearly proportioal to the umber of bay regios. Key words: Estrada Idex; Bezeoid Hydrocarbos; Molecular Graph; Spectrum (of Graph). 1. Itroductio I this paper we are cocered with a molecular structure-descriptor of bezeoid hydrocarbos that we refer to as the Estrada idex. Itisdefiedasfollows. Let G be the molecular graph of a bezeoid hydrocarbo [1 5]. Let ad m be respectively the umber of vertices ad edges of G. The the formula of the uderlyig hydrocarbo is C H 3 ad must be eve [3 5]. The eigevalues λ 1 λ 2...λ of the adjacecy matrix of G are said to be the eigevalues of G adtoform the spectrum of G. These will be labelled so that λ 1 λ 2 λ. The basic properties of graph eigevalues ca be foud i [6]. The Estrada idex is defied as EE = EE(G)= e λ i. (1) Although itroduced quite recetly [7] the Estrada idex has already foud umerous applicatios. It was used to quatify the degree of foldig of logchai molecules especially proteis [7 9]. Aother fully urelated applicatio of EE was put forward by Estrada ad Rodríguez-Velázquez [10 11]. They showed that EE provides a measure of the average cetrality of complex (commuicatio social metabolic etc.) etworks. I additio to this i a recet work [12] a coectio betwee EE ad the cocept of exteded atomic brachig was suggested. Util ow oly some straightforward mathematical properties of the Estrada idex were established [ ] but its depedece o molecular structure has ot bee ivestigated. The preset paper is aimed at cotributig towards fillig this gap. I what follows we use the fuctios hyperbolic cosie ad hyperbolic sie defied as usual as cosh(x)= ex + e x ad sih(x)= ex e x 2 2 respectively. Our startig poits are the well kow relatios for the eigevalues of bezeoid graphs [2 6] λ i + λ +1 i fori = (2) kow as the pairig theorem [ ] ad /2 (λ i ) 2 = m. (3) Because of (2) half of the eigevalues of a bezeoid graph G are positive (or zero) ad the other half egative (or zero) implyig [11] /2 EE(G)=2 cosh(λ i ) / 07 / $ c 2007 Verlag der Zeitschrift für Naturforschug Tübige

2 I. Gutma ad S. Radeković Estrada Idex of Bezeoid Hydrocarbos A McClellad-Type Boud for the Estrada Idex The method by which a boud EE for EE is deduced i this sectio is fully aalogous to a method used log time ago [16] for estimatig the total π- electro eergy E. I the otatio specified above for a bezeoid graph G /2 E = E(G)=2 λ i. (4) For a fixed value of m i. e. assumig that coditio (3) is obeyed a extremal value E of E is obtaied by employig the Lagrage multiplier techique: A auxiliary fuctio F is costructed as /2 F := 2 λ i α ad the coditio ) (λ i ) 2 m ( /2 F imposed for all k = 12.../2. This leads to 2 2αλ k i. e. λ k = α which combied with (3) yields λ k = for k = 12.../2 ad substituted back ito (4) results i E =. This is just the famous McClellad upper boud for total π-electro eergy [17 19]. Applyig a aalogous reasoig we costruct the auxiliary fuctio /2 FF := 2 cosh(λ i ) α ad impose ) (λ i ) 2 m ( /2 FF for k = 12.../2. This results i 2sih(λ k ) 2αλ k. It is easily see that for α > 0 the equatio sih(x) α x has a sigle positive-valued solutio. Deote it by.theα = sih( )/. Now from λ k = for k = 12.../2 adrelatio (3) we readily obtai = ad therefore EE = cosh ( ). (5) Note that / is the average vertex degree of the graph G. Therefore for bezeoid graphs / > 2 ad thus > 2. Curiously however i cotrast to E which is a upper boud for the total π-electro eergy [17] the estimate EE formula (5) is a lower boud for the Estrada idex. To see this we examie the Hessia matrix H(FF) of the fuctio FF. Because of FF = 2sih(λ k ) 2αλ k oe has 2 FF λk 2 = 2cosh(λ k ) 2α ad 2 FF λ k for k k. Therefore H(FF) is a diagoal matrix whose all diagoal elemets are equal to 2cosh( ) 2α i. e. 2 cosh( ) 2 sih(). Now 2cosh( ) 2 sih() = ( 1)e +( + 1)e which for > 2 is evidetly positive-valued. Thus all eigevalues of H(FF) are positive-valued ad cosequetly the extremal value EE is a miimum. We thus proved:

3 256 I. Gutma ad S. Radeković Estrada Idex of Bezeoid Hydrocarbos EE = a 1 EE + a 2 (6) a 1 = ± a 2 = 0.64 ± 0.08 with a remarkably high correlatio coefficiet I the sample examied the average relative error of the approximatio (6) is 0.19% ad the maximal observed relative error 0.83%. Oe evidet coclusio from the above result is that the gross part of the Estrada idices of bezeoid systems is determied by the parameters ad m.iother words the Estrada idices of bezeoid isomers differ very little. I the subsequet sectio we examie these small differeces of the EE values of isomeric bezeoids ad try to see which is the mai structural feature that is resposible for them. 4. O Estrada Idices of Bezeoid Isomers Expadig the fuctio e x ito a power series ad usig the defiitio (1) of the Estrada idex oe readily arrives at [7 10] Fig. 1. The Estrada idices (EE) of the 106 bezeoid hydrocarbos from [22] plotted versus their lower boud EE accordig to (5). For details see text. Theorem 1. The Estrada idex of a bezeoid hydrocarbo with carbo atoms ad m carbo-carbo bods is always greater tha cosh( /). 3. A (m m)-type Approximatio for the Estrada Idex The McClellad upper boud E for a total π- electro eergy E provides a excellet approximate formula for E of cojugated molecules [17] of the form E a 1 E +a 2. I fact i the case of bezeoid hydrocarbos this formula with a ad a 2.45 happes to be the best (m)-type approximatio for E [20 21]. I view of this we examied how well a expressio of the form a 1 EE + a 2 would approximate the Estrada idex. The quality of this approximatio is see i Figure 1. Ideed the correlatio betwee EE ad EE is almost perfectly liear. Least-squares fittig usig the stadard data set of 106 Kekuléa bezeoid hydrocarbos from [22] (same as employed i [20 21] ad elsewhere [23 24]) yields the regressio lie EE(G)= k 0 M k (G) k! where M k is the k-th spectral momet of the molecular graph G M k = M k (G)= (λ i ) k. For all graphs M 0 = M 1 ad M 2 = [2 6]. For all bipartite graphs (ad thus also for the molecular graphs of bezeoid hydrocarbos) M k for odd k. We thus have EE(G)=+m M M M 8 + (7) which implies EE(G) + m M M 6. (8) The depedece of the first few eve spectral momets of bezeoid hydrocarbos o the molecular structure is kow [25 27]. I particular M 4 (G)=18m 12 M 6 (G)=158m b where b is the umber of bay regios [ ]. Whe these are substituted back ito (7) ad (8) we obtai EE(G)= 1 (1418m b) higher order terms

4 I. Gutma ad S. Radeković Estrada Idex of Bezeoid Hydrocarbos 257 Table 1. Statistical data for correlatios of the form EE = a 1 b + a 2 for sets of bezeoid isomers with carbo atoms ad m carbo-carbo bods; b is the umber of bay regios. All sets cosidered cotai all possible isomers equal to N.I. The respective correlatio coefficiet is R. m N.I. a 1 a 2 R ± ± ± ± ± ± ± ± ± ± Fig. 2. The Estrada idices (EE) of the 36 bezeoid isomers with the formula C 26 H 16 plotted versus their umber of bay regios (b). For details see Table 1 ad text. i. e. EE(G) 1 (1418m b). (9) 720 [1] N. Triajstić Chemical Graph Theory CRC Boca Rato [2] I. Gutma ad O. E. Polasky Mathematical Cocepts i Orgaic Chemistry Spriger-Verlag Berli [3] I. Gutma ad S. J. Cyvi Itroductio to the Theory of Bezeoid Hydrocarbos Spriger-Verlag Berli [4] M. Zader Z. Naturforsch. 45a 1041 (1990). [5] M. Zader Topics Curr. Chem (1990). [6] D. Cvetković M. Doob ad H. Sachs Spectra of Graphs Theory ad Applicatio Academic Press New York [7] E. Estrada Chem. Phys. Lett (2000). [8] E. Estrada Bioiformatics (2002). [9] E. Estrada Proteis (2004). [10] E. Estrada ad J. A. Rodríguez-Velázquez Phys. Rev. 71E (2005). Accordig to the approximatio (9) withi classes of isomeric bezeoids the Estrada idex is a icreasig liear fuctio of the parameter b ad the slope of the respective lie is (almost) idepedet of ad m ad (early) equal to 1/ That this is ideed the case is see from Fig. 2 ad from the data give i Table Cocludig Remarks We deem to have established the mai structural features of bezeoid hydrocarbos that determie the value of their Estrada idices. These are first of all the parameters ad m that are capable of reproducig some 99.8% of EE. The(m)-type approximatio a 1 EE +a 2 with EE beig give by (5) is maybe ot the best possible but is remarkably accurate. The quality of the liear relatio betwee EE ad EE is illustrated by Figure 1. Ayway the Estrada idices of bezeoid isomers (i. e. species havig equal values of ad m) vary oly to a very limited extet. The mai structural feature ifluecig these variatios is the umber b of bay regios. Withi sets of bezeoid isomers EE is a icreasig liear fuctio of b. The slope of this fuctio is practically idepedet of ad m (as see from the data for a 1 i Table 1). It is close yet ot equal to the slope predicted by meas of a trucated expasio of EE i terms of spectral momets. We dare to coclude that the structure depedece of the Estrada idices of bezeoid hydrocarbos is ow almost completely uderstood. [11] E. Estrada ad J. A. Rodríguez-Velázquez Phys. Rev. 72E (2005). [12] E. Estrada J. A. Rodríguez-Velázquez ad M. Radić It. J. Quatum Chem (2006). [13] I. Gutma E. Estrada ad J. A. Rodríguez-Velázquez Croat. Chem. Acta 79 (i press). [14] I. Gutma Z. Naturforsch. 39a 152 (1984). [15] R. B. Mallio ad D. H. Rouvray J. Math. Chem. 5 1 (1990). [16] I. Gutma MATCH Commu. Math. Comput. Chem (1983). [17] B. J. McClellad J. Chem. Phys (1971). [18] A. Graovac I. Gutma P. E. Joh D. Vidović ad I. Vlah Z. Naturforsch. 56a 307 (2001). [19] H. Fripertiger I. Gutma A. Kerber A. Kohert ad D. Vidović Z. Naturforsch. 56a 342 (2001). [20] I. Gutma Topics Curr. Chem (1992).

5 258 I. Gutma ad S. Radeković Estrada Idex of Bezeoid Hydrocarbos [21] I. Gutma ad T. Soldatović MATCH Commu. Math. Comput. Chem (2001). [22] R. Zahradik ad J. Pacir HMO Eergy Characteristics Pleum Press New York [23] J. Cioslowski ad I. Gutma Z. Naturforsch. 41a 861 (1986). [24] I. Gutma J. H. Koole V. Moulto M. Parac T. Soldatović ad D. Vidović Z.Naturforsch.55a 507 (2000). [25] J. Cioslowski Z. Naturforsch. 40a 1167 (1985). [26] G. G. Hall Theor. Chim. Acta (1986). [27] S. Marković ad I. Gutma J. Mol. Struct. (Theochem.) (1991).