Wiener Index: Formulas for Non-homeomorphic Graphs

FORMULAS FOR NON-HOMEOMORPHIC GRAPHS 4552B" 22 +2E" 22 + F' =F 3 = 5 (27)B" 22 = B 3 =32C 22 + E" 22 = E 3 = –6 (28)B + 22 = B 3 = 3 (29)2A 22 +2B 22 +2D 22 + E 22 +2F 22 + G 22 = 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 +4C 22 +4E" 22 + F' = –1 (32)4A 22 +4B 22 +8B' 22 +4D 22 +2E 22 +4F 22 + G 22 = 0 (33)Finally, by choosing o = n = 1 again a simple chain results, consisting ofthree parts (see Appendix, Eq. (A4)). Comparison of coefficients k 2 , k, kx andthe constant yields the following equations, respectively:B 22 + B' 22 + D 22 = –6 (34)B 22 + B' 22 + E 22 + E' 22 + F 22 = 11 (35)C 22 + E" 22 = – 12 (36)2A 22 +2B' 22 +2D 22 + E' 22 +2F 22 + G 22 = –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 often