Wiener Index: Formulas for Non-homeomorphic Graphs

FORMULAS FOR NON-HOMEOMORPHIC GRAPHS 457The 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 A 2 , B 2 , D 2 , E 2 and F 2 . A »starlike« graph consistingof two side-chains – or stings – is in fact a chain consisting of two »subchains«a and b. The Wiener formula of W derived for N-chains is: 1andW =(N 3 – N)/6N = a + b – 1(A1)(A2)(a + b –1) 3 –(a + b –1)= a 3 + b 3 +3a 2 b +3ab 2 –3a 2 –3b 2 –6ab +2a +2b. (A3)Therefore, A 2 =1,B 2 =3,D 2 =3,E 2 =–6andF 2 = 2, in accordance withformulas (13) – (17). Similarly,(a + b + c –2) 3 –(a + b + c –2)=(a 3 + b 3 + c 3 )+3(a 2 b + ab 2 + a 2 c + ac 2 ++ b 2 c + bc 2 )+6abc –6(a 2 + b 2 + c 2 ) – 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. Gutman, Croat. Chem. Acta (1998) (to appear),(d) A. A. Dobrynin and I. Gutman, Graph Theory (New York) 28 (1995) 21–23,(e) A. Graovac and T. Pisanski, J. Math. Chem. 8 (1991) 53–62,(f) O. Ivanciuc and A. Balaban, Commun. Math. Chem. (MATCH) 30 (1994)141–152, P. John, Commun. Math. Chem. 32 (1995) 207–219, M. Randi}, Chem.Phys. Lett. 211 (1993) 478–483, D. Rouvray and B. C. Crafford, S. Afr. J. Sci. 72(1976) 47–51, S. S. Tratch, M. I. Stankevich, and N. S. Zefirov, J. Comput. Chem.11 (1990) 899–908, D. J. Klein and M. Randi}, J. Math. Chem. 12 (1993) 81–95, Z.Mihali}, S. Nikoli}, and N. Trinajsti}, J. Chem. Inf. Comput. Sci. 32 (1992) 28–37.7. R. C. Entringer, D. E. Jackson, and D. A. Snyder, Czech Math. J. 8 (1984) 1–21.