Abstract
We present a novel connectivity index for (molecular) graphs, called sum-connectivity index and give several basic properties for this index, especially lower and upper bounds in terms of graph (structural) invariants. It appears that this and the original Randić connectivity index that we call product-connectivity index are highly intercorrelated molecular descriptors, the value of the correlation coefficient being 0.991 for trees representing lower alkanes. We determine the unique tree with fixed numbers of vertices and pendant vertices with the minimum value of the sum-connectivity index, and trees with the minimum, second minimum and third minimum, and the maximum, second maximum and third maximum values of this index. Additionally, we discuss the properties of this novel connectivity index for a class of trees representing acyclic hydrocarbons.
Similar content being viewed by others
References
Randić M.: On characterization of molecular branching. J. Am. Chem. Soc. 97, 6609–6615 (1975)
Kier L.B., Hall L.H.: Molecular Connectivity in Chemistry and Drug Research. Academic Press, New York (1976)
Kier L.B., Hall L.H.: Molecular Connectivity in Structure-Activity Analysis. Research Studies Press/Wiley, Letchworth/New York (1986)
R. Todeschini, V. Consonni, Handbook of Molecular Descriptors (Wiley-VCH, Weinheim, 2000)
Pogliani L.: From molecular connectivity indices to semiempirical connectivity terms: Recent trends in graph theoretical descriptors. Chem. Rev. 100, 3827–3858 (2000)
Garcia-Domenech R., Galvez J., de Julian-Ortiz J.V., Pogliani L.: Some new trends in chemical graph theory. Chem. Rev. 108, 1127–1169 (2008)
Randić M.: On history of the Randić index and emerging hostility toward chemical graph theory. MATCH Commun. Math. Comput. Chem. 59, 5–124 (2008)
Kier L.B., Hall L.H., Murray W.J., Randić M.: Molecular-connectivity I: Relationship to nonspecific local anesthesia. J. Pharm. Sci. 64, 1971–1974 (1975)
Bollobás B., Erdös P.: Graphs of extremal weights. Ars Comb. 50, 225–233 (1998)
Bollobás B., Erdös P., Sarkar A.: Extremal graphs for weights. Discr. Math. 200, 5–19 (1999)
I. Gutman, Chemical graph theory—The mathematical connection, in Advances in Quantum Chemistry, vol. 51, ed. by J.R. Sabin, E.J. Brändas (Elsevier, Amsterdam, 2006), pp. 125–138
Rada J., Uzcátegui C.: Randić ordering of chemical trees. Discr. Appl. Math. 150, 232–250 (2005)
X. Li, I. Gutman, Mathematical Aspects of Randić-Type Molecular Structure Descriptors (University of Kragujevac, Kragujevac, 2006)
Pepper R., Klein D.J.: Some theorems about the Randić connectivity index. MATCH Commun. Math. Comput. Chem. 58, 359–364 (2007)
Li X., Shi Y.: A survey on the Randić index. MATCH Commun. Math. Comput. Chem. 59, 127–156 (2008)
I. Gutman, B. Furtula (eds.), Recent Results in the Theory of Randić Index (University of Kragujevac, Kragujevac, 2008)
Randić M.: The connectivity index 25 years after. J. Mol. Graph. Modell. 20, 19–35 (2001)
Bonchev D.: Overall connectivity—a next generation molecular connectivity. J. Mol. Graph. Modell. 20, 65–75 (2001)
Estrada E.: Edge adjacency relationships and a novel topological index related to molecular volume. J. Chem. Inform. Comput. Sci. 35, 31–33 (1995)
Nikolić S., Trinajstić N., Baučić I.: Comparison between the vertex- and edge-connectivity indices for benzenoid hydrocarbons. J. Chem. Inform. Comput. Sci. 38, 42–46 (1998)
Amić D., Davidovic-Amić D., Bešlo D., Lučić B., Trinajstić N., Nikolić S.: The vertex-connectivity index revisited. J. Chem. Inform. Comput. Sci. 38, 819–822 (1998)
Stankevich V., Skvortsova M.I., Zefirov N.S.: On a quantum chemical interpretation of molecular connectivity indices for conjugated hydrocarbons. J. Mol. Struct. (THEOCHEM) 342, 173–179 (1995)
Galvez J.: On a topological interpretation of electronic and vibrational molecular energies. J. Mol. Struct. (THEOCHEM) 429, 255–264 (1998)
Estrada E.: Physicochemical interpretation of molecular connectivity indices. J. Phys. Chem. A 106, 9085–9091 (2002)
Klein D.J., Palacios J.L., Randić M., Trinajstić N.: Random walks and chemical graph theory. J. Chem. Inform. Comput. Sci. 44, 1521–1525 (2004)
B. Zhou, D. Vukičević, On general Randić and general zeroth-order Randić indices. MATCH Commun. Math. Comput. Chem. 62, in press (2009)
Gutman I., Trinajstić N.: Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons. Chem. Phys. Lett. 17, 535–538 (1972)
Gutman I., Ruščić B., Trinajstić N., Wilcox C.F. Jr.: Graph theory and molecular orbitals. XII. Acyclic polyenes. J. Phys. Chem. 62, 3399–3405 (1975)
S.C. Basak, G.D. Grunwald, G.J. Niemi, Use of graph-theoretic geometric molecular descriptors in structure-activity relationships, in From Chemical Topology to Three-Dimensional Geometry, ed. by A.T. Balaban (Plenum Press, New York, 1997), pp. 73–116
S.C. Basak, B.D. Gute, G.D. Grunwald, A hierarchical approach to the development of QSAR models using topological, geometrical and quantum chemical parameters, in Topological Indices and Related Descriptors in QSAR and QSPR, ed. by J. Devillers, A.T. Balaban (Gordon & Breach, Amsterdam, 1999), pp. 675–696
A.T. Balaban, I. Motoc, D. Bonchev, O. Mekenyan, Topological indices for structure–activity correlations, in Steric Effects in Drug Design, ed. by M. Charton, I. Motoc (Topics in Current Chemistry 114, Springer, Berlin, 1983), pp. 21–55
Gutman I., Das K.C.: The first Zagreb index 30 years after. MATCH Commun. Math. Comput. Chem. 50, 83–92 (2004)
Das K.C., Gutman I.: Some properties of the second Zagreb index. MATCH Commun. Math. Comput. Chem. 52, 103–112 (2004)
Zhou B.: Zagreb indices. MATCH Commun. Math. Comput. Chem. 50, 113–118 (2004)
Vukičević D., Trinajstić N.: On the discriminatory power of the Zagreb indices for molecular graphs. MATCH Commun. Math. Comput. Chem. 53, 111–138 (2005)
Zhou B., Gutman I.: Further properties of Zagreb indices. MATCH Commun. Math. Comput. Chem. 54, 233–239 (2005)
Hansen P., Vukičević D.: Comparing the Zagreb indices. Croat. Chem. Acta 80, 165–168 (2007)
Nikolić S., Kovačević G., Miličević A., Trinajstić N.: The Zagreb indices 30 years after. Croat. Chem. Acta 76, 113–124 (2003)
Vukičević D., Trinajstić N.: Modified Zagreb M2 index—Comparison with Randić connectivity index of benzenoid systems. Croat. Chem. Acta 76, 183–187 (2003)
Miličević A., Nikolić S.: On variable Zagreb indices. Croat. Chem. Acta 77, 97–101 (2004)
F. Harary, Graph Theory, 2nd printing (Addison-Wesley, Reading, MA, 1971)
Wilson R.J.: Introduction to Graph Theory. Oliver and Boyd, Edinburgh (1972)
Trinajstić N.: Chemical Graph Theory, 2nd edn. CRC Press, Boca Raton (1992)
Ivanciuc O., Ivanciuc T., Cabrol-Bass D., Balaban A.T.: Evaluation in quantitative structure–property relationship models of structural descriptors derived from information-theory operators. J. Chem. Inform. Comput. Sci. 40, 631–643 (2000)
de Caen D.: An upper bound on the sum of squares of degrees in a graph. Discr. Math. 185, 245–248 (1998)
Li J., Pan Y.: de Caen’s inequality and bounds on the largest Laplacian eigenvalue of a graph. Linear Algebr. Appl. 328, 153–160 (2001)
Das K.C.: Maximizing the sum of the squares of the degrees of a graph. Discr. Math. 285, 57–66 (2004)
Zhou B., Stevanović D.: A note on Zagreb indices. MATCH Commun. Math. Comput. Chem. 56, 571–578 (2006)
Caporossi G., Gutman I., Hansen P., Pavlović L.: Graphs with maximum connectivity index. Comput. Biol. Chem. 27, 85–90 (2003)
Hansen P., Mélot H.: Variable neighborhood search for extremal graphs, 6. Analyzing bounds for the connectivity index. J. Chem. Inform. Comput. Sci. 43, 1–14 (2003)
Gutman I., Polanski O.E.: Mathematical Concepts in Organic Chemistry. Springer, Berlin (1986)
Gutman I., Miljković O., Caporossi G., Hansen P.: Alkanes with small and large Randić connectivity indices. Chem. Phys. Lett. 306, 366–372 (1999)
Gutman I., Miljković O.: Molecules with smallest connectivity indices. MATCH Commun. Math. Comput. Chem. 41, 57–70 (2000)
Gutman I.: Molecular graphs with minimal and maximal Randić indices. Croat. Chem. Acta 75, 357–369 (2002)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Zhou, B., Trinajstić, N. On a novel connectivity index. J Math Chem 46, 1252–1270 (2009). https://doi.org/10.1007/s10910-008-9515-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10910-008-9515-z