Cluj and Related Polynomials Applied in Correlating Studies

Transcription

1 864 J. Chem. Inf. Model. 2007, 47, Cluj and Related Polynomials Applied in Correlating Studies Mirea V. Diudea,*, A. E. Vizitiu, and Dušanka Janežič Faulty of Chemistry and Chemial Engineering, Babes-Bolyai University, Arany Janos 11, Cluj, Romania, and National Institute of Chemistry, Hajdrihova 19, SI-1000 Ljubljana, Slovenia Reeived November 1, 2006 A ounting polynomial P(G,x) is a desription of a graph property P(G) in terms of a sequene of numbers so that the exponents express the extent of its partitions while the oeffiients are related to the frequeny of the ourrene of partitions. Basi definitions and properties of Cluj ounting polynomials CJ(G,x) and their relation with Ω(G,x) and NΩ(G,x) polynomials are presented. Analytial relations for the alulation of suh polynomials and their single-number desriptors in some lasses of planar polyhexes are derived. The ability of these desriptors to predit the boiling point, hromatographi retention index, and resonane energy for some planar polyhex ompounds, as well as the toxiity of a set of dibenzofurans, is demonstrated. 1. INTRODUCTION A graph G(V,E), in whih V is the set of verties and E the set of its edges, an be desribed by a onnetion table, a sequene of numbers, a single number (alled a topologial index), a matrix, or a polynomial. Quantum hemistry was the first field that used the polynomial desription of a moleular graph. In the early Hükel theory, the roots of the most studied harateristi polynomial 1,2 P(G,x) ) det[xi - A(G)] (1) are related to the π-eletron energy levels of the moleular orbitals in onjugated hydroarbons. In eq 1, I is the unit matrix of a pertinent order and A the adjaeny matrix of graph G. The harateristi polynomial is involved in the topologial resonane energy, the topologial effet on moleular orbitals, the aromati sextet theory, and the Kekulé struture ount. 1-5 An extension of relation 1 was made by Hosoya et al. 6 and others 7,8 who hanged the adjaeny matrix by the distane matrix and, alternatively, by any square topologial matrix. The polynomial oeffiients, m(g,k), as presented in the equation P(G,x) ) k m(g,k)xk (2) an be alulated from the graph by a method whih makes use of the Sahs graphs, whih are subgraphs of G. Equation 2 was derived independently by Sahs, Harary, Milić, Spialter, and Hosoya. 2 This method is useful with small graphs, but in larger graphs, the numeri methods of linear algebra, suh as the reursive algorithms of Le Verier, Frame, or Fadeev, are more effiient. 9,10 Dediated to Professor Nenad Trinajstić on the oasion of his 70th birthday. * Corresponding author diudea@hem.ubbluj.ro. Babes-Bolyai University. National Institute of Chemistry, Slovenia. Equation 2 is a general expression of a ounting polynomial, written as a sequene of numbers, with the exponents showing the extent of partitions p(g), p(g) ) P(G) ofa graph property P(G) while the oeffiients m(g,k) are related to the ourrene of partitions of extent k. Counting polynomials were introdued in the mathematial hemistry literature by Hosoya 11,12 with his Z-ounting (independent edge sets) and the distane degree (initially alled Wiener and later Hosoya) 13,14 polynomials. Their roots and oeffiients are used for the haraterization of the topologial nature of hydroarbons. Hosoya and Yamaguhi also proposed the sextet polynomial 15,16 for ounting the resonant rings in a benzenoid moleule. The sextet polynomial is important in onnetion with the Clar aromati sextets, 17,18 whih are expeted to stabilize the aromati moleules. The independene polynomial ounts the seletions of k-independent verties of G. Other related graph polynomials are the king, olor, and star polynomials If one ounts sets of mutually adjaent verties instead of the sets of independent verties, one obtains the lique polynomial. 24 Polynomials are the subjet of an extensive review. 1 For some distane-related properties, the polynomial oeffiientsarealulablefromthelayerandshellmatries, whih an be built aording to the vertex distane partitions of a graph and are alulable by the TOPOCLUJ software pakage. 30 This paper is foused on the mathematial aspets of the Cluj polynomials in relation to Ω-type polynomials, whih ould be a promise in the mathematial desription of topology of various lasses of moleules, with possible use in orrelating studies. The definitions of Cluj matries are provided in the next setion; the Cluj polynomials are defined in the third setion, and the properties of the Cluj and two related ounting polynomials are presented in the fourth setion. The final setion presents several examples of the utility of desriptors derived from Cluj and related polynomials in the predition of some physiohemial and biologial properties of polyyli strutures /i600482j CCC: $ Amerian Chemial Soiety Published on Web 04/18/2007

2 CLUJ AND RELATED POLYNOMIALS J. Chem. Inf. Model., Vol. 47, No. 3, Chart 1 Figure 1. Diphenylene DPH and its numbering. In graphs ontaining rings, more than one path ould join the pair (i,j), resulting in more than one fragment related to i (with respet to j and a given path p). The entries in the Cluj matrix are taken, by definition, as the maximum ardinality among all suh fragments: [UCJ] i,j ) max p In trees, due to the unique nature of paths joining any two verties, CJ i,j,p represents the set of paths going to j through i. In this way, the path p(i,j) is haraterized by a single endpoint, whih is suffiient for the unsymmetri matrix UCJ. When the path p belongs to the set of distanes DI(G), the suffix DI is added to the name of matrix, as in UCJDI. When path p belongs to the set of detours DE(G), the suffix is DE. The Cluj matries are defined in any graph and, exept for some symmetri graphs, are unsymmetri and an be symmetrized by the Hadamard multipliation with their transposes If the matries alulated from edges (i.e., on adjaent vertex pairs) are required, the matries alulated from paths must be multiplied by the adjaeny matrix A (whih has the nondiagonal entries of 1 if the verties are joined by an edge and, otherwise, zero) The basi properties and appliations of the above matries and derived desriptors have been presented elsewhere Examples of Cluj matries, alulated on distane or detour, are given in Chart 1; the numbering is shown in Figure CLUJ POLYNOMIALS CJ i,j,p (4) SM p ) UM (UM) T (5) SM e ) SM p A (6) The Cluj polynomials are defined on the basis of Cluj matries as 2. CLUJ MATRICES A Cluj fragment, 1,31-34 CJ i,j,p, ollets verties V lying loser to i than to j, the endpoints of a path p(i,j). Suh a fragment ollets the vertex proximity of i against any vertex j, joined by the path p, with the distanes measured in the subgraph D (G-p), as shown in the following equation: CJ i,j,p ) {V V V(G); D (G -p) (i,v) < D (G -p) (j,v)} (3) CJ(G,x) ) k m(g,k)xk (7) They ount the vertex proximity of the vertex i with respet to any vertex j in G, joined to i by an edge {p e,i } (the Clujedge polynomials) or by a path {p p,i } (the Cluj-path polynomials), taken as the shortest (distane DI) or the longest (detour DE) paths. In eq 7, the oeffiients m(g,k) are alulated from the entries of Cluj matries. The summation runs up to the maximum k ) {p} in G.

3 866 J. Chem. Inf. Model., Vol. 47, No. 3, 2007 DIUDEA ET AL. Figure 2. Edge ut proedure for alulating Cluj polynomial CJDI e in diphenylene DPH. Table 1. Cluj CJDI e and Related Polynomials in Phenylenes PHE h; h ) Number of Hexagons in Moleule CJDI e (PHE h,x) ) 4 h-1 x 6k k)1 h x 6k+3 + 4hx 3h ; D1 x)1 ) 48h 2-12h ) 6h(8h - 2) ) Ve k)0 Ω(PHE h,x) ) (3h - 1)x 2 + x 2h ; D1 x)1 ) e ) 8h - 2 CI(PHE h ) ) 4(3h - 1)(5h - 2) Ω(PHE h, x) x)1 ) 3h ) V/2 NΩ(PHE h,x) ) 2hx (6h-2) + 2(3h - 1)x 4(2h-1) ; D1 x)1 ) 4(3h - 1)(5h - 2) Table 2. Cluj CJDI e and Related Polynomials in Spiranes SP h; h ) Number of Hexagons in Moleule CJDI e (SP 1,3,x) ) (8 + 2h)x h-1 k)2 x (5k-2) + (8 + 2h)x 5h-2 ; D1 x)1 ) ev CJDI e (SP 1,4,,x) ) 12 h k)1 x (5k-2) ; D1 x)1 ) ev CJDI e (SP 1,3;1,4,x); D1 x)1 ) ev ) 6h(5h + 1) Ω(SP 1,3;1,4,x) ) 3hx 2 ; D1 x)1 ) e ) 6h and V ) 5h + 1 NΩ(SP 1,3;1,4,x) ) 6hx (e-2) ) 6hx (6h-2) ; D1 x)1 ) 12h(3h - 1) Table 3. Cluj CJDI e and Related Polynomials in Pyrenes PYR p/h (d-condensed Pyrenes, See #10, Table 14); p ) PYR Units; h ) Number of Hexagons in Moleule CJDI e (PYR p,x) ) 4px p-1 k)0 x (5+14k) + 2(2p + 1)x 7p p-1 k)0 x (11+14k) + 4px 14p-1 ; D1 x)1 ) ev ev ) 2(18p + 1)(7p + 1) ) D1[CJDI e (PYR p,x)] x)1 Ω(PYR p,x) ) 2px 2 + 4px 3 + x 2p+1 ; D1 x)1 ) e ) 18p + 1; V ) 2(7p + 1) NΩ(PYR p,x) ) 4px (18p-1) + 12px (18p-2) + (2p + 1)x 16p ; D1 x)1 ) 4p(80p - 3) CJDI e (PYR h,x) ) hx h/4-1 k)0 x (5+14k) + (h + 2)x 7h/ h/4-1 k)0 x (11+14k) + hx 7h/2-1 ; D1 x)1 ) ev ev ) (9h/2 + 1)(7h/2 + 2) ) D1[CJDI e (PYR h,x)] x)1 Ω(PYR h,x) ) h/2x 2 + hx 3 + x h/2+1 ; D1 x)1 ) e ) 9h/2 + 1; V ) 7h/2 + 2 NΩ(PYR h,x) ) hx (9h/2-1) + 3hx (9h/2-2) + (h/2 + 1)x 4h ; D1 x)1 ) h(20h - 3) The Cluj polynomials published previously 26 referred to some partitions of the Cluj matries given by the layer/shell matries and provide no diret interpretation of the ounting ontent. The polynomial oeffiients are ounted from the Cluj matries by the TOPOCLUJ software program 30 and also by a simple routine for olleting the entries in the unsymmetri matries UCJ. In the ase of the CJDI e polynomial, an orthogonal edge-utting proedure an be used, as shown in Figure 2. The same proedure has been used by Gutman and Klavžar to alulate the Szeged index of polyhex graphs. 35 In this paper, V)V(G) ) V(G) and e ) e(g) ) E(G) refer to the ardinality of the vertex and edge sets, respetively. Sine the Cluj matries are unsymmetri, two polynomials an be written, one with respet to the endpoint i (and to the first triangle of the UCJDI e matrix) and the other one referring to the endpoint j of the edge e(i,j) (and to the seond triangle of that matrix). They depend, however, on the

4 CLUJ AND RELATED POLYNOMIALS J. Chem. Inf. Model., Vol. 47, No. 3, Table 4. Cluj CJDI e and Related Polynomials in Coronenes CORr; r ) Number of Hexagon Rows around the Central Hexagon CJDI e (COR r,x) ) r k)0 6(r + k + 2)x (k+1)(2r+k+3) + r-1 k)0 6(r + k + 2)x 6(r+1)2-(k+1)(2r+k+3) ; D1 x)1 ) ev ev ) 18(r + 1) 3 (3r + 2) ) D1[CJDI e (COR r,x)] x)1 ; ev ) 18(r) 3 (3r - 1) ) D1[CJDI e (COR r,x)] x)1 V ) 6(r + 1) 2 ; f ) 3r(r + 1) + 2; e ) 3(r + 1)(3r + 2) Ω(COR r,x) ) 6 (r+1) k)2 x (r+k) + 3x 2(r+1) ; D1 x)1 ) 3(r + 1)(3r + 2) ) e NΩ(COR r,x) ) r+1 k)2 6(r + k)x [e-(r+k)] + 6(r + 1)x [e-2(r+1)] ; D1 x)1 ) (r + 1)(81r r r + 24) numbering, and only their sum polynomial is an invariant (see Figure 2). Only this last polynomial will be onsidered in the following disussion. 4. PROPERTIES OF THE CLUJ AND RELATED POLYNOMIALS Among the properties of ounting polynomials, the value in x ) 1 and the first derivative in x ) 1 are the most important. In the ase of the CJDI e polynomial, the value in x ) 1isP x)1 ) 2e. It is evident, sine every edge is visited twie. The first derivative, in x ) 1, gives the meaning of the topologial property olleted by a matrix or a polynomial. In this ase, the following theorem holds: Theorem 1. In a bipartite graph, the sum of all edgeounted Vertex proximities equals the produt V e of the number of Verties and edges in G. Demonstration. In a bipartite, planar graph, in whih orthogonal edge uts are permitted, for every edge e(i,j) E(G), there is a lear separation of vertex proximities {p e,i } and {p e,j } of its endpoints. If we denote the ardinalities of the above sets by p e,i and p e,j, then in a bipartite graph It follows that, for all edges, e E(G), the total of edgeounted vertex proximities p e equals the produt V e, thus demonstrating the theorem. An orthogonal or an elementary edge ut of a (polyyli) bipartite planar graph G is a straight line segment, orthogonal to and passing through the enters of some edges of the graph, interseting the perimeter of G exatly two times, so that at least one polygon lies between these two intersetion points. 1,35 Trees also allow elementary edge uts. In the orthogonal edge-ut proedure used to alulate the CJDI e index (i.e., the half-sum of all entries in the matrix CJDI e ), the total of edge-ounted vertex proximities p e ) CJDI e (G,x); D1 x)1 is alulated as p e ) p e,i + p e,j )V (8) m(g,) (p e,i + p e,j ) )V m(g,) )V e (9) where the oeffiients m(g,) are related to the ourrene of edge uts of extent and m(g,) ) e(g). The above theorem an be extended to 3D bipartite moleular strutures, although the separation of the proximities is less obvious in this ase. Numerial results provided by the Cluj matrix support this extension. Corollary to Theorem 1. In bipartite graphs, there are no equidistant Verties with respet to the two endpoints of any edge. The Cluj matrix ounts the verties lying loser to eah of the endpoints of any edge and leaves the equidistant verties unounted. Beause of relation 8, it follows that, in bipartite graphs, all verties are ounted and no equidistant Verties exist. Next, for all of the edges in G, one obtains the total of vertex proximities, p e, equal to V e. This is the main result provided by the Cluj matrix/polynomial. The vertex proximity alulation ould be of interest in alulating the bond polarity and moleular dipole moments. In this respet, weighted moleular graphs must be used. It may be realled that equidistant verties are also not ounted in alulating the Szeged index, a topologial index related to the Wiener index, 39 whih ounts all the shortest distanes in a graph. Theorem 2. In a tree graph G, the sum of all path-ounted Vertex proximities is twie the sum of all distanes in G or twie the Wiener index W: p p ) CJDI p (G,x); D1 x)1 ) 2W. Demonstration. The olumn sums in the UCJDI p matrix equal the olumn sums in the matrix of distanes, while the row sums in the UCJDI p matrix are idential to those in the Wiener matrix. 1 It is known that the half sum of entries in these matries ounts all the distanes in a tree graph, or the Wiener index. Sine the first derivative of the CJDI p polynomial is the sum of all entries in UCJDI p, it follows that p p ) CJDI p (G,x); D1 x)1 ) 2W, thus demonstrating the theorem. In graphs ontaining rings, the Cluj index CJDI p however is different from both the Wiener and Szeged indies. 1 Formulas for alulating the Cluj CJDI e and related polynomials (see below), in phenylenes, spiranes, pyrenes, and oronenes, are given in Tables 1-4. Two related polynomials are inluded in the above tables. One is the Omega polynomial, Ω(G,x), proposed by Diudea 40 for ounting the orthogonal edge uts, and the seond one is the non-omega NΩ(G,x) polynomial, whih is somewhat omplementary to the Omega polynomial. It has been shown 40,41 that the edges forming an edge-ut strip are loally odistant, or equidistant and topologially parallel. Let m(g,) denote the ourrene of the edge ut of length (i.e., the number of edges ut off) in G. In a bipartite, planar graph, the two polynomials are defined as Ω(G,x) ) m(g,) x (10) NΩ(G,x) ) m(g,) x (e-) (11)

5 868 J. Chem. Inf. Model., Vol. 47, No. 3, 2007 DIUDEA ET AL. Table 5. Polyhex Moleules Tested for Correlation vs BP and I hr with Ω(G,x) as an Identifier Their D1 x)1 s give the total number of equidistant and nonequidistant edges versus eah edge in G Ω(G,x)D1 x)1 ) e ) E(G) (12) NΩ(G,x)D1 x)1 ) PI(G) (13) where PI(G) is Khadikar s topologial index. 42 Two indies have been defined on the basis of the Omega polynomial. The first, CI, is derived from the first and seond derivatives, in x ) 1, as CI(G) ) [Ω(G,x)D1] 2 - [Ω(G,x)D1 + Ω(G,x)D2] x)1 (14) The seond desriptor an be alulated from all possible

6 CLUJ AND RELATED POLYNOMIALS J. Chem. Inf. Model., Vol. 47, No. 3, Table 6. Properties and Topologial Indies of the Moleular Strutures from Table 5 G BP I CHR UCJDI p UCJDE p I Ω CI SCJDI p SCJDE p SCJDI e SCJDE e Table 7. Correlating Ability of Desriptors from Table 6 in Mono and Bivariate Regression against BP monovariate bivariate indies R CV% indies R CV% 1 UCJDE p SCJDI p, CI CI UCJDI p, CI SCJDI e SCJDI e,ci UCJDI p SCJDE e,ci SCJDE p UCJDE p,ci SCJDI p SCJDE p,ci SCJDE e I Ω Table 8. Correlating Ability of Desriptors from Table 6 in Monoand Bivariate Regression against I hr monovariate bivariate indies R CV% indies R CV% 1 CI SCJDI p,ci UCJDE p SCJDI e,ci UCJDI p UCJDI p,ci SCJDI e SCJDE e,ci SCJDI p UCJDE p,ci SCJDE p SCJDE p,ci SCJDE e derivatives Dn, in x ) 1, and normalized to the first derivative (whih equals the number of edges in G) Theorem 3. In a bipartite planar graph, CI and PI indies are idential. Demonstration. From eqs 11 and 13, writing m(g,) simply as m, the PI index an be alulated as PI(G) ) I Ω (G) ) [1/Ω(G,x)D1] n m(e - ) ) e m - [Ω(G,x)Dn] 1/n x)1 (15) m 2 ) e 2 - m 2 (16) CI may always be alulated using the relation developed by the expansion of eq 14 as CI(G) ) (m) 2 - [m + m( - 1)] ) ) e 2 - In the above, the following relation holds: e(g) ) m 2 ) PI(G) (17) m ) Ω(G,x)D1 x)1 ) NΩ(G,x) x)1 (18) However, eq 11 is not always valid, so that CI is, in general, different from PI, exept in the ase of bipartite planar graphs. Equation 16 was proposed by John et al. 43 for omputing PI in benzenoid hydroarbon graphs. In bipartite graphs whih an be embedded in surfaes of g > 0, for example, in toroidal polyhexes, relation 11 is more obsure, notwithstanding CI ) PI. In the ase of the torus T(6,3)H: 8,12 Ω(G,x) ) 12x 4 + 4x 24, e ) 144, CI ) , NΩ(G,x) ) 96x x 136, and PI ) Aording to eq 11, the polynomial would be NΩ(G,x) ) 96x x 140 ; this gives the same PI index value. Currently, no generalization of this ase ould be found. The CJDI(G,x) and NΩ(G,x) polynomials desribe, in the same manner, olletions of nonequidistant subgraphs (verties and edges, respetively), addition being the most simple and natural operation. Appliations of these desriptors are desribed below. 5. CORRELATING ABILITY OF CLUJ AND RELATED DESCRIPTORS To test the orrelating ability of Cluj desriptors, we seleted different types of planar polyhex strutures, dibenzofurans, oronenes, and pyrenes Boiling Point and Chromatographi Retention Index of Polyhexes. The physiohemial properties estimated for the strutures in Table 5 are the boiling

7 870 J. Chem. Inf. Model., Vol. 47, No. 3, 2007 DIUDEA ET AL. Table 9. Interorrelating Matrix of the Desriptors from Table 6 CI I Ω SCJDE e UCJDE p SCJDE p SCJDI e UCJDI p SCJDI p CI I Ω SCJDE e UCJDE p SCJDE p SCJDI e UCJDI p SCJDI p 1 Table 10. Struture and AhR Binding Affinity Data for Chlorinated Dibenzofurans struture substituent toxiity (pec 50) 1 2-Cl Cl Cl ,3-diCl ,6-diCl ,8-diCl ,2,7-trCl ,3,6-trCl ,3,8-trCl ,3,4-trCl ,3,8-trCl ,2,3,6-teCl ,2,3,7-teCl ,2,4,8-teCl ,3,4,6-teCl ,3,4,7-teCl ,3,4,8-teCl ,3,6,8-teCl ,3,7,8-teCl ,2,3,4,8-peCl ,2,3,7,8-peCl ,2,3,7,9-peCl ,2,4,6,7-peCl ,2,4,6,8-peCl ,2,4,7,8-peCl ,2,4,7,9-peCl ,3,4,7,8-peCl ,3,4,7,8-peCl ,3,4,7,9-peCl ,2,3,4,7,8-heCl ,2,3,6,7,8-heCl ,2,4,6,7,8-heCl ,3,4,6,7,8-heCl dibenzofuran Figure 3. Dibenzofurans; predited vs observed toxiity. Figure 4. Plot of resonane energy against that alulated by Cluj desriptors (Table 16). point (BP) and hromatographi retention index (I hr ). 44,45 The goal of this study was not to find the best model for a given moleular property but to obtain information about the possibility of using the Cluj and related desriptors in orrelating studies. Aordingly, only monoand bivariate regressions were onsidered. Table 5 also inludes Ω(G,x) as an identifier of the planar polyhex strutures. The properties and topologial indies of the set of strutures in Table 5 are listed in Table 6. Statistis of the test are listed in Tables 7 and 8. Both indies derived from the Omega polynomial show the same degeneray. It is as expeted, sine both indies are alulated on this polynomial, whih is also degenerate, exatly on the same strutures. Figure 5. Plot of resonane energy against that alulated by eq 21. Also as expeted, the indies belonging to the same lass are highly interorrelated (here, in bold, with the threshold 0.97). The high orrelation of CI with the Cluj desriptors is quite surprising. By ontrast, I Ω is pratially orthogonal to all of these desriptors (Table 9). In monovariate regression, we an onsider a good orrelation to be one with CV% < 10. In the ase of BP, all but I Ω show a signifiant orrelation. However, only that of

8 CLUJ AND RELATED POLYNOMIALS J. Chem. Inf. Model., Vol. 47, No. 3, Table 11. Partial Charges (Ch), Regression Coeffiients ( i), and Ct DBF in Dibenzofurans moleule Ch 2 Ch 3 Ch 4 Ch 5 Ch 7 Ch 9 Ch 10 Ch 11 Ch 13 Ct DBF a i b a Global desriptor of partial harges in DBF set. b Interept term. Table 12. Data for the Training Set of Dibenzofurans moleule Ct DBF C (LM_W_CJDIe _ SGI) toxiity units UCJDE p and CI is aeptable. In the ase of I hr, four desriptors exeeded the perent of ovariane (CV%) threshold of 10. In bivariate regression, the threshold for good orrelation was taken as CV% < 5. In the ase of BP, three pairs of desriptors fulfilled this ondition, while in the ase of I hr, only one pair of desriptors sueeded in surpassing the threshold. This ensures a Pearson orrelation oeffiient R > 0.97, a threshold also providing a good orrelation. In all the bivariate orrelations, CI is the pair; no pair of exlusively Cluj desriptors ould provide a good orrelation. This Table 13. Data for the Testing Set of Dibenzofurans moleule Ct DBF C (LM_W_CJDIe _ SGI) (pec 50) toxiity toxiity (pec 50) ald demonstrates that CI, along with the Cluj desriptors, is a good desriptor in orrelating studies. The CV% and the Pearson oeffiient R are good indiators of the quality of the regression equation. No linear size effet was inferred from the above study, but a lear struture-property orrelation is manifested. Our results are at least as good as those reported earlier by Randić. 44,45 Table 9 shows an interorrelating matrix of the desriptors from Table 6.

9 872 J. Chem. Inf. Model., Vol. 47, No. 3, 2007 DIUDEA ET AL. Table 14. Moleular Strutures of Pyrenes Table 16. Data for the Pyrenes from Table 14 struture UCJDI p SCJDI e RE (ev) RE ald Toxiity of Dibenzofurans. Dibenzofurans are industrially important and are known for their wide distribution, resistane to biologial and hemial degradations, high toxiity, and bioaumulation. 46,47 Some of them ause toxi effets after binding to an intraellular ytosoli reeptor, the aryl hydroarbon reeptor (AhR). 48,49 To desribe the struture of dibenzofurans 50 in Table 10, a very simple eletroni desriptor based on partial harges was generated and orrelated on an ad ho basis with the property studied. Partial harges, alulated with the PM3 Hamiltonian, 51 and the oeffiients of autoorrelation 52,53 are listed in Table 11. The last olumn ontains the global desriptor Ct DBF, whih is a linear ombination of the loal partial harges fitted to the autoorrelation equation. We assoiated this global desriptor with an index of entrality, 1 C (LM_W_CJDIe _SGI) alulated on the Cluj matrix CJDI e with a weighting sheme using Sanderson group eletronegativities, SGI, 28 to aount for the hemial nature of atoms in moleules. A subset of 16 dibenzofurans (Table 12) was randomly seleted as the training set, and in a leave-halfout approah, 15 strutures (Table 13) were used as the Table 15. Coronene-Like Strutures

10 CLUJ AND RELATED POLYNOMIALS J. Chem. Inf. Model., Vol. 47, No. 3, Table 17. Data for the Set of Coronenes from Table 16 struture I HOMO (ev) UCJDip RE (ev) RE ald (ev) residual test set. The following results were obtained: (i) In the training set, a orrelation of over 0.8 was obtained (ii) In the test set, only a small derease in orrelation Toxiity ) Ch DBF C (LM_W_CJDe _SGI) (19) R 2 ) 0.888, n ) 16, F ) , s ) was seen (R v2 ) 0.854) and there was also a small inrease in the standard error of the estimate (s ) 0.574). This indiates that the model is valid. A plot of the observed versus alulated toxiity of dibenzofurans is given in Figure 3. Sine the biologial response is multionditional, our results are statistially signifiant, and the desriptor weighted by the hemial nature of atoms funtions orretly in the above model. The above results an be ompared with the best one reported by Basak: 50 R v2 ) 0.862, in four variables Resonane Energy of Polyhexes. Two sets of polyhex strutures have been seleted. 54 The first onsists of pyrene derivatives (Table 14); the other omprises polyhexes related to oronene (Table 15), and the modeled property was the resonane energy. Compared to the timeonsuming ab initio density funtional theory or even semiempirial alulations, some easily alulated desriptors suh as the Cluj desriptors ould be attrative in prediting, with a lower omputational ost and adequate auray, the resonane energy of suh moleules. Even a learly nonlinear size dependene of the quantum parameter was reognized. In this ontext, the good orrelations reported here are only a rough measure of the size effet inferred here. No diret omparable studies on these sets are available. Table 16 lists data for the pyreni strutures from Table 14. Only Cluj desriptors have been used. The orrelation is shown in Figure 4. In the ase of the strutures formally derived from oronene (Tables 15 and 17), an initial orrelation was made on 14 strutures. In addition to the Cluj desriptors, we used the I Ω index, alulated on the Omega polynomial, and the value of the highest oupied moleular orbital (HOMO) orbital, alulated at the PM3 level of theory. The orrelation is given in eq 20 RE ) I HOMO UCJDI p (20) n ) 14, R 2 ) , F ) , s ) Compound 3 (Table 15) had a residual value, res ) , twie larger than the standard error (s), and this struture (or its RE value) was onsidered to be an outlier and was removed from the set. The equation alulated using the remaining 13 moleules revealed better statistis (eq 21), thus proving that moleule #3 is an outlier RE ) I Ω HOMO UCJDI p n ) 13, R 2 ) , F ) , s ) (21) The plot of resonane energy against that alulated by eq 21 is given in Figure 5. The topologial desriptors used in this work were alulated with our original software. 30,55,56 6. CONCLUSIONS A novel lass of ounting polynomials, alled Cluj polynomials, has been proposed on the basis of the previously desribed Cluj matries. It was shown that the polynomial oeffiients an be alulated from the Cluj matries or by means of orthogonal edge uts in the ase of the CJDI e version. Analytial formulas for alulating the Cluj and related polynomials in phenylenes, spiranes, pyrenes, and oronenes have been developed. Basi definitions and properties of the Cluj matries and orresponding polynomials are given. The meaning of the Cluj desriptors, as vertex proximity desriptors, is demonstrated. For CJDI e, it was shown that, in bipartite graphs, the sum of all edge-ounted vertex proximities equals the produt V e of the number of verties and edges in the graph. In trees, the sum of all path-ounted vertex proximities is twie the Wiener index. The relationship of Cluj polynomials CJDI(G,x) with Ω(G,x) and NΩ(G,x) polynomials is desribed. The desriptors derived from the Cluj and related polynomials were used to predit the boiling point, hromatographi retention index, and resonane energy of sets of planar polyhex strutures. The use of vertex proximity alulations in evaluating the bond polarity and moleular dipole moments was suggested. The toxiity of polyhlorinated dibenzofurans was modeled by using an autoorrelated syntheti index derived from partial harges of atoms and added to a Cluj desriptor. The Cluj and related polynomial desriptors proved to be useful

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