The Wiener index (W) is a graph invariant. Because of its importance, various methods – numerical approaches and analytical formulas – have been proposed to compute W. Earlier formulas of W were derived for homeomorphic structures, polymers, and stars. The formulas are polynomials. In this paper, a method is proposed that allows calculating the coefficients of the polynomials in sets of non-homeomorphic structures

István Lukovits

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Lukovits, István

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ISSN-0011-1643CCA-2504 Original Scientific PaperWiener Index: Formulas for Non-homeomorphicGraphsIstván LukovitsChemical Research Center, Hungarian Academy of Sciences,H–1525, Budapest, P. O. Box 17, HungaryReceived January 20, 1998; revised April 2, 1998; accepted May 22, 1998The Wiener index (W) is a graph invariant. Because of its impor-tance, various methods – numerical approaches and analytical for-mulas – have been proposed to compute W. Earlier formulas of Wwere derived for homeomorphic structures, polymers, and stars.The formulas are polynomials. In this paper, a method is proposedthat allows calculating the coefficients of the polynomials in sets ofnon-homeomorphic structures.INTRODUCTIONThe Wiener-number or Wiener-index (W) is a graph-invariant.1 W isequal to the sum of distances (i.e. the number of bonds on the shortest path)di,j between all pairs of vertices i and j:W = di ji j, . (1)As an example, consider the computation of the Wiener-index of iso-butane (Figure 1, hydrogens will not be considered throughout this pa-per). W (isobutane) = d1,2 + d1,3 + d1,4 + d2,3 + d2,4 + d3,4 = 1 + 1 + 1 + 2 + 2 + 2 = 9.CROATICA CHEMICA ACTA CCACAA 71 (3) 449¿458 (1998)Figure 1. Hydrogen suppressed graph of isobutane. The numbers indicate vertices.W is one of the most freqnently applied graph theoretical invariants. Whas been used to explain the variation in boiling points, molar volumes, re-fractive indices, heats of isomerization and heats of vaporization of alkanes.1Later heats of formation, atomization, isomerization, and vaporization aswell as density, critical pressure, surface tension, viscosity, melting points,partition coefficients, chromotographic retention indices, and stability ofcrystal lattices of various kinds of molecules were related2,3 to W. Besideschemistry, the concept of the Wiener number has been used in electricalengineering4 and mathematics.5 Quite a few graph invariants related to Whave been proposed, too.6In addition to Eq. (1), analytical formulas have been proposed for W.Wiener derived a formula for chains1 (i.e. for acyclic structures having twoendpoints), Entringer et al.7 and Bonchev et al.8 proposed formulas for sim-ple cycles, Trinajsti} et al. obtained formulas for polymers9 and Gutman de-rived formulas for various types of acyclic graphs.10 A formula was also de-rived for fused bicyclic structures.11 Canfield et al. proposed a recursionmethod to obtain appropriate formulas for any acyclic structure.12 Each ofthese formulas is valid for a given set of homeomorphic structures, polymersor stars. Two graphs are homeomorphic if both can be obtained from thesame graph by a sequence of subdivisions of lines. For example any two cy-cles are homeomorphic, any two chains are homeomorphic and the star de-picted in Figure 1 is homeomorphic with the structure depicted in Figure 2.In this paper, a method will be proposed that allows to obtain formulas for abroader set of structures.DERIVATION OF FORMULASExpressions »vertex« and »atom«, »chemical structure« and »graph«,»bond« and »edge«, »valence« and »degree« are synonymous words used inchemistry and graph theory, respectively, and will be used interchangeablyhereafter.Consider a »starlike« graph (Figure 2) that has a single branching ver-tex, and three side-chains k, m, and n, where k, m and n denote both thenumber of vertices of the side chain and the side-chains themselves. A star-like graph (Figure 2) is homeomorphic with a star which has a branching450 I. LUKOVITSFigure 2. Example of a »starlike« graph containing three chains. k = 2, m = 3, and n= 4. The branching vertex belongs to all three chains.vertex of the same degree (Figure 1). The branching atom belongs to allside-chains. Example: isobutane (Figure 1) consists of three side chains, andeach side chain contains two vertices. Instead of »side-chain«, we shall usethe expression »string« to denote a chainlike subgraph starting with an end-point or a branching vertex and ending with and endpoint or a branchingvertex. An atom between these two vertices (if any) is bivalent.The Wiener index of a starlike graph containing strings k, m and n (Fig-ure 2) may be calculated by using the following formula:2,3W = (k3 + m3 + n3) + 3(k2m + k2n + m2n + km2 + kn2 + mn2) –– 6(k2 + m2 + n2) – 6(km + kn + mn) + 5 (k + m + n)/6. (2)For isobutane with k = m = n = 2, we obtain:W = (23 + 23 + 23) + 3(222 + 222 + 222 + 222 + 222 + 222) – 6(22 + 22 +22) – 6(22 + 22 + 22) + 5(2 + 2 + 2)/6 = (24 + 144 – 72 – 72 + 30)/6 = 9, thesame value that was obtained by using Eq. (1). Eq. (2) remains valid if anyof the side chains disappears, i. e. the size of the corresponding string – e.g.n – is equal to 1. Let us use Eq. (2) to obtain W for propane, then k = 3,m = n = 1, and W = (33 + 13 + 13) + 3(321 + 321 + 121 + 312 + 312 +112) – 6(32 + 12 + 12) – 6(31 + 31 +11) + 5(3 + 1 + 1)/6 = (29 + 78 – 66 – 42 + 25)/6 = 4, in accordance with thevalue that could be obtained for the (hydrogen suppressed) graph of propane.We want to derive a similar formula for a starlike graph containing fourstrings k, m, n and o (Figure 3) and the formula should have the followingform:W = A4(k3 + m3 + n3 + o3) + B4(k2m + k2n + k2o + m2n + m2o + n2o ++ km2 + kn2 + ko2 + mn2 + mo2 + no2) + D4(k2 + m2 +n2+ o2) ++ E4(km + kn + ko + mn + mo+ no) + F4(k + m + n + o)/6 (3)FORMULAS FOR NON-HOMEOMORPHIC GRAPHS 451Figure 3. Example of a »starlike« graph containing four chains. The branching ver-tex belongs to all four chains, k = 2, m = 3, n = 4, and o = 5.where coefficients A4, B4, D4, E4 and F4, have to be determined. The sub-scripts indicate that the starlike graph contains four side-chains (strings).For similar reasons, the corresponding coefficients in Eq. (2) will be denotedby A3, B3, D3, E3 and F3, where A3 = 1, B3 = 3, D3 = –6, E3 = –6 and F3 = 5.Note that Eq. (3) cannot contain other third order terms, like kmn, aconstant, fourth or higher order terms.11Let as suppose that coefficients A4, B4, D4, E4 and F4 are already known,and we want to use Eq. (3) to derive Eq. (2). Assume that o = 1, and substi-tuting o = 1 into Eq. (3), we obtain for the constant value in Eq. (2) whichequals zero:A4 + D4 + F4 = 0. (4)Similarly, the sum of the coefficients of the k2 term in Eq. (3) is equal to –6,the coefficient of the k2 term in Eq. (2):B4 + D4 = D3 = –6. (5)By comparing the coefficients of k in Eqs. (2) and (3), we obtain:B4 + E4 + F4 = F3 = 5. (6)The present procedure cannot generate an equation for A3 or B3, since thecoefficients of the third order terms k3 and k2m, A4 and B4, respectively, arenot affected by setting o = 1. Therefore A4 = A3 = 1 and B4 = B3 = 3, and theset of equations (4)–(6) can be solved to obtain D4 = –9, E4 = –6 and F4 = 8.From this, we can write down the equation for starlike structures with foursidechains:W = (k3 + m3 + n3 + o3) + 3(k2m + k2n + k2o + m2n + m2o + n2o ++ km2 + kn2 + ko2 + mn2 + mo2 + no2) – 9(k2 + m2 + n2 + o2) –– 6(km + kn + ko + mn + mo + no) + 8 (k + m + n + o)/6. (7)Example: consider structure of 3,3-dimethylpentane (Figure 4), With k =m = 2, n = o = 3 and W = (8 + 8 + 27 + 27) + 3(8 + 12 + 12 + 12 + 12 + 27 + 8+ 18 + 18 + 18 + 18 + 27) – 9(4 + 4 + 9 + 9) – 6(4 + 6 + 6 + 6 + 6 + 9) + 8(2 + 2 +3 + 3)/6 = (70 + 570 – 234 – 222 + 80)/6 = 44.The procedure can be extended for starlike structures eith n sidechains.Again An = 1 and Bn = 3. In full analogy with equations (4)–(6), we maywrite:An + Dn + Fn = 0 (8)Bn + Dn = Dn–1 (9)452 I. LUKOVITSBn + En + Fn = Fn–1. (10)From Eq. (9) we obtain:Dn = Dn–1 – 3. (11)By expressing Fn from Eq. (8) and using Eq. (11), we obtain:Fn = –Dn – 1 = 2 – Dn–1. (12)Finally, by expressing En from Eq. (10) and from Eqs. (11) and (12), weobtain:En = Fn–1 – Fn – 3 = Fn–1 + Dn–1 – 2 – 3 = Dn–1 – Dn–2 – 3 == Dn–2 – Dn–2 – 6 = –6. (13)meaning that En = –6, irrespective of the value of n. Finally, by taking intoaccount the numerical values of coefficients A3, B3, D3, E3 and F3 in Eq. (2),we obtain:An = 1 (14)Bn = 3 (15)Dn = –3(n –1) (16)En = –6 (17)Fn = 3n – 4. (18)Coefficients An, Bn, Dn, En and Fn may be used to obtain W for any starlikegraph in terms of the sizes of strings. The coefficients are still valid if n = 2(see Appendix).The present procedure can be applied to obtain the coefficients of anygraph consisting of two »starlike« subgraphs. The simplest case is depictedin Figure 5. The polynomial we want to derive for this class of homeomor-phic graphs should have the following form:FORMULAS FOR NON-HOMEOMORPHIC GRAPHS 453Figure 4. Hydrogen suppressed graph of 3,3-dimethylpentane.W = (A22(k3 + o3 + m3 + n3) + A'22x3 + B22(k2o + ko2 + m2n + mn2) ++ B'22(k2m + k2n + o2m + o2n + km2 + kn2 + om2 + on2) + B"22(k2x ++ o2x + m2x + n2x) + B+22(kx2 + ox2 + mx2 + nx2) + C22(kxm + kxn ++ oxm + oxn) + D22(k2 + o2 + m2 + n2) + D'22x2 + E22(ko + mn) ++ E'22(km + om + kn + on) + E"22(kx + ox + mx + nx) ++ F22(k + o + m + n) + F'22x + G22/6 (19)where A22, A'22, B22, B'22, B"22, B+22, C22, D22, D'22, E22, E'22, E"22, F22, F'22and G22 are the coefficients to be determined. The subscripts indicate thattwo side-chains emerge from both ends of the central chain x.Observe that for x = 1 the resulting graph is a starlike graph havingfour strings (Figure 3). Comparison with coefficients of terms k2, k, km andthe constant in Eq. (7) yields:B"22 + D22 = D4 = – 9 (20)B+22 + E"22 + F22 = F4 = 8 (21)C22 + E'22 = E4 = –6 (22)A'22 + D'22 + F'22 + G22 = 0. (23)By choosing m = n = 1, a starlike graph with three side-chains (Figure 3)results. Comparison with coefficients of terms k2, k, x2, x, k2x, kx, kx2 andthe constant in Eq. (7) yields:2B'22 + D22 = D3 = –6 (24)2B'22 + 2E'22 + F22 = F3 = 5 (25)2B+22 + D'22 = D3 = –6 (26)454 I. LUKOVITSFigure 5. Example of a graph containing two starlike subgraphs. The branchingvertices belong to three incident chains. k = 2, j = 4, m = 3, n = 5, and (the stringlinking the two branching vertices) x = 5.2B"22 + 2E"22 + F' = F3 = 5 (27)B"22 = B3 = 32C22 + E"22 = E3 = –6 (28)B+22 = B3 = 3 (29)2A22 + 2B22 + 2D22 + E22 + 2F22 + G22 = 0. (30)By choosing k = o = m = n = 1, a simple chain results, and the chain formuladoes not contain second and zero order (constant) terms (see Appendix, Eq.(A1)), whereas the coefficient of x (the first order term) is equal to –1.4B+22 + D' = 0 (31)4B"22 + 4C22 + 4E"22 + F' = –1 (32)4A22 + 4B22 + 8B'22 + 4D22 + 2E22 + 4F22 + G22 = 0 (33)Finally, by choosing o = n = 1 again a simple chain results, consisting ofthree parts (see Appendix, Eq. (A4)). Comparison of coefficients k2, k, kx andthe constant yields the following equations, respectively:B22 + B'22 + D22 = –6 (34)B22 + B'22 + E22 + E'22 + F22 = 11 (35)C22 + E"22 = – 12 (36)2A22 + 2B'22 + 2D22 + E'22 + 2F22 + G22 = –6 . (37)Solution of the system of linear equations (20)–(37) yields the coefficientsof Eq. (19). Numerical values of the coefficients are listed in Table I.A similar procedure can be used to calculate the coefficients for a graphcontaining two branching atoms of degree four (Figure 6), and in general forany graph containing two branching atoms with a degree of k + 1 (Table I).DISCUSSIONThe formulas presented in this paper may be used to solve the problemof »inverse quantitative structure – activity relationship«,13,14 at least forstarlike graphs or graphs consisting of two starlike subgraphs. Quantitativestructure-activity relationships (QSAR) denote statistical investigations inwhich the numerical values of some activity (or property) of the moleculesare related to a number (or numbers) depending on the structure of themolecule. The independent variables appearing in QSARs are quite oftenFORMULAS FOR NON-HOMEOMORPHIC GRAPHS 455graph invariants, particularly W is an important variable.1–3 As a result ofQSAR investigations, the activity of a molecule is expressed as a function ofgraph invariants characterizing its structure. The aim of the inverse QSARproblem is to find out which molecules have the desired property. In order tosolve this problem, we have to generate all graphs possessing a definitevalue of W (or any other graph invariant).In such a procedure, the minimal and maximal7,15 values of vertices forwhich graphs with a definite W may exist have to be determined first. Then,formulas derived in this work can be used to generate all starlike graphs orgraphs consisting of two starlike subgraphs containing an allowed numberof vertices and to select all those structures which correspond to the fixedvalue of W. The time needed to perform this task depends on the number ofstrings to be considered. The number of strings will necessarily be limited.As a result of this procedure, the same graph might be considered severaltimes, and a method to avoid this difficulty has to be found.456 I. LUKOVITSTABLE ICoefficients of polynomials related to graphs containing twostarlike subgraphs (Figure 5)Coefficient n = 2 n = 3 n = kAnn = A'nnBnn = B'nn = B"nn = B+nnCnnDnn = D'nnEnnE'nnE"nnFnnF'nnGnn136–12–6–12–182335–24136–18–6–12–243571–54136–6k–6–12–6(k + 1)12k – 16k2 + 6k – 1–6k2Figure 6. Scheme of a graph containing two starlike subgraphs. The branching verti-ces belong to four incident chains.The generation procedure described above is restricted to starlikegraphs or structures for which formulas have already been dervied.11 Thesolution of the inverse QSAR problem applicable to any three has still to besolved.APPENDIXCalculate coefficients A2, B2, D2, E2 and F2. A »starlike« graph consistingof two side-chains – or stings – is in fact a chain consisting of two »sub-chains« a and b. The Wiener formula of W derived for N-chains is:1W = (N3 – N)/6 (A1)andN = a + b – 1 (A2)(a + b – 1)3 – (a + b – 1) = a3 + b3 + 3a2b + 3ab2 – 3a2 – 3b2 – 6ab + 2a + 2b. (A3)Therefore, A2 = 1, B2 = 3, D2 = 3, E2 = –6 and F2 = 2, in accordance withformulas (13) – (17). Similarly,(a + b + c – 2)3 – (a + b + c – 2) = (a3 + b3 + c3) + 3(a2b + ab2 + a2c + ac2 ++ b2c + bc2) + 6abc – 6(a2 + b2 + c2) – 12(ab + ac + bc) + 11(a + b + c) – 6. (A4)REFERENCES1. H. Wiener, J. Am. Chem. Soc. 69 (1947) 2636–2638.2. I. Gutman, Y. N. Yeh, S. L. Lee, and Y. L. Luo, Indian J. Chem. 32A (1993)651–661.3. S. Nikoli}, N. Trinajsti}, and Z. Mihali}, Croat. Chem. Acta 68 (1995) 105–129.4. P. G. Doyle and J. L. Snell, Random Walks and Electric Networks, Math. Assoc.,Washington, 1984.5. J. K. Doyle and J. E. Graver, Discrete Math. 17 (1977) 147–154.6. (a) D. Ami} and N. Trinajsti}, Croat. Chem. Acta 68 (1995) 53–62,(b) M. V. Diudea, J. Chem. Inf. Comput. Sci. 36 (1996) 535–540,(c) M. V. Diudea and I. 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(Theochem) 285 (1993) 137–142.11. I. Lukovits,, J. Chem. Inf. Comput. Sci. 31 (1991) 503–507.12. E. R. Canfield, R. W. Robinson, and D. H. Rouvray, J. Comput. Chem. 6 (1985)598–609.13. L. B. Kier, L. H. Hall, and J. W. Frazer, J. Chem. Inf. Comput. Sci. 33 (1993)143–147.14. M. I. Skvortsova, I. I. Baskin, O. L. Slovokhotova, V. A. Palyulin, and N. S. Zefirov,J. Chem. Inf. Comput. Sci. 33 (1993) 630–634.15. D. Bonchev and N. Trinajsti}, J. Chem. Phys. 67 (1977) 4517–4533.SA@ETAKWienerov indeks: formule za nehomeomorfne grafoveIstván LukovitsWienerov indeks (W) invarijanta je grafa. Zbog njegove va`nosti, predlo`eni surazli~iti postupci za njegovo izra~unavanje, uklju~uju}i numeri~ke pristupe i anali-ti~ke formule. Ve} izvedene formule za W prikazane su specijalno za homeomorfnestrukture, polimere i zvijezde, u polinomnom obliku. U ovom radu predlo`en je po-stupak za izra~unavanje koeficijenata polinoma za skupove nehomeomorfnih struk-tura.458 I. LUKOVITS

Wiener Index: Formulas for Non-homeomorphic Graphs

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