A Unified Approach to Extremal Cacti for Different Indices

Many chemical indices have been invented in theoretical chemistry, such as Wiener index, Merrifield-Simmons index, Hosoya index, spectral radius and Randic index, etc. The extremal trees and unicyclic graphs for these chemical indices are interested in existing literature. Let G be a molecular graph (called a cacti), which all of blocks of G are either edges or cycles. Denote G (n, r) the set of cacti of order n and with r cycles. Obviously, G (n, 0) is the set of all trees and G (n, 1) is the set of all unicyclic graphs. In this paper, we present a unified approach to the extremal cactus, which have the same or very similar structures, for Wiener index, Merrifield-Simmons index, Hosoya index and spectral radius. From our results, we can derive some known results.

[1]  M. Randic.  Characterization of molecular branching , 1975 .

[2]  LU Mei,et al.  On the Randić index of cacti , 2006 .

[3]  H.B. Walikar,et al.  On the Eigenvalues of a Graph , 2003, Electron. Notes Discret. Math..

[4]  Miroslav Petrović,et al.  On the index of cactuses with n vertices , 2006 .

[5]  Zlatko Mihalić,et al.  A graph-theoretical approach to structure-property relationships , 1992 .

[6]  H. Wiener.  Vapor pressure-temperature relationships among the branched paraffin hydrocarbons. , 1948, The Journal of physical and colloid chemistry.

[7]  Aimei YUa,et al.  A Kind of Graphs with Minimal Hosoya Indices and Maximal Merrifield-Simmons Indices∗ , .

[8]  Feng Tian,et al.  Extremal Catacondensed Benzenoids , 2003 .

[9]  R. Merrifield,et al.  Topological methods in chemistry , 1989 .

[10]  I. Gutman.  Extremal hexagonal chains , 1993 .

[11]  I. Gutman,et al.  Mathematical Concepts in Organic Chemistry , 1986 .

[12]  L. Hall,et al.  Molecular connectivity in chemistry and drug research , 1976 .

[13]  Harold Ellsworth Tinnappel.  On the topological index , 1952 .

[14]  H. Wiener.  Structural determination of paraffin boiling points. , 1947, Journal of the American Chemical Society.

[15]  S. Unger.  Molecular Connectivity in Structure–activity Analysis , 1987 .

[16]  Jinlong Shu,et al.  A sharp upper bound on the largest eigenvalue of the Laplacian matrix of a graph , 2002 .

[17]  Lemi Türker.  Contemplation on the Hosoya indices , 2003 .

[18]  H. Hosoya.  Topological Index. A Newly Proposed Quantity Characterizing the Topological Nature of Structural Isomers of Saturated Hydrocarbons , 1971 .

[19]  Roberto Todeschini,et al.  Handbook of Molecular Descriptors , 2002 .

[20]  Ivan Gutman,et al.  Algebraic Connections between Topological Indices , 1998, J. Chem. Inf. Comput. Sci..