Use of the Szeged index and the revised Szeged index for measuring network bipartivity

We have revisited the Szeged index (Sz) and the revised Szeged index (Sz^*), both of which represent a generalization of the Wiener number to cyclic structures. Unexpectedly we found that the quotient of the two indices offers a novel measure for characterization of the degree of bipartivity of networks, that is, offers a measure of the departure of a network, or a graph, from bipartite networks or bipartite graphs, respectively. This is because the two indices assume the same values for bipartite graphs and different values for non-bipartite graphs. We have proposed therefore the quotient Sz/Sz^* as a measure of bipartivity. In this note we report on some properties of the revised Szeged index and the quotient Sz/Sz^* illustrated on a number of smaller graphs as models of networks.

[1]  Tomaz Pisanski,et al.  Weights on Edges of Chemical Graphs Determined by Paths , 1994, J. Chem. Inf. Comput. Sci..

[2]  S. Trifunović,et al.  The synthesis and characterization of facial and meridional isomers of uns-cis-(ethylenediamine-N-N'-di-3-propionato) cobalt(III) complexes with S-lysine and S-histidine , 2002 .

[3]  Patrick W. Fowler.  Resistance distances in fullerene graphs , 2002 .

[4]  Milan Randic,et al.  Novel Shape Descriptors for Molecular Graphs , 2001, J. Chem. Inf. Comput. Sci..

[5]  Gordon F. Royle,et al.  Algebraic Graph Theory , 2001, Graduate texts in mathematics.

[6]  Danail Bonchev,et al.  Graph Theoretical Approaches to Chemical Reactivity , 1994 .

[7]  István Lukovits,et al.  Extensions of the Wiener Number , 1996, J. Chem. Inf. Comput. Sci..

[8]  Janez Zerovnik.  Szeged Index of Symmetric Graphs , 1999, J. Chem. Inf. Comput. Sci..

[9]  Ante Graovac,et al.  On the Wiener index of a graph , 1991 .

[10]  Milan Randi.  On Complexity of Transitive Graphs Representing Degenerate Rearrangements , 2001 .

[11]  L. Lovász,et al.  On the eigenvalues of trees , 1973 .

[12]  Dejan Plavsic,et al.  On the Relation between the P'/P Index and the Wiener Number , 1996, J. Chem. Inf. Comput. Sci..

[13]  Milan Randic,et al.  Wiener Matrix Invariants , 1994, Journal of chemical information and computer sciences.

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

[15]  Janez Zerovnik,et al.  Corroborating a modification of the Wiener index , 2002 .

[16]  M. Petkovšek,et al.  Counting disconnected structures : Chemical trees, fullerenes, I-graphs, and others , 2005 .

[17]  J. A. Rodríguez-Velázquez,et al.  Spectral measures of bipartivity in complex networks. , 2005, Physical review. E, Statistical, nonlinear, and soft matter physics.

[18]  Milan Randić.  On generalization of wiener index for cyclic structures , 2002 .

[19]  Milan Randic,et al.  Symmetry properties of chemical graphs. IV. Rearrangement of tetragonal-pyramidal complexes , 1982 .

[20]  Tomaz Pisanski,et al.  Growth in products of graphs , 2002, Australas. J Comb..

[21]  Tomislav Doslic,et al.  Computing the bipartite edge frustration of fullerene graphs , 2007, Discret. Appl. Math..

[22]  Onn Chan,et al.  Wiener Number as an Immanant of the Laplacian of Molecular Graphs , 1997, J. Chem. Inf. Comput. Sci..

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

[24]  Dejan Plavšić,et al.  On the Relation between W '/W Index, Hyper-Wiener Index, and Wiener Number , 2000, J. Chem. Inf. Comput. Sci..

[25]  B. Mohar,et al.  How to compute the Wiener index of a graph , 1988 .

[26]  Janez Žerovnik.  COMPUTING THE SZEGED INDEX , 1996 .

[27]  Dejan Plavsic,et al.  On the Calculation of the Molecular Descriptor '/ , 1998, J. Chem. Inf. Comput. Sci..

[28]  M. Randic.  Novel molecular descriptor for structure—property studies , 1993 .

[29]  Subhash C. Basak,et al.  On ordering of folded structures , 2000 .

[30]  Milan Randić,et al.  Generalized molecular descriptors , 1991 .

[31]  Petter Holme,et al.  Network bipartivity. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[33]  Tomaz Pisanski,et al.  Edge-contributions of some topological indices and arboreality of molecular graphs , 2009, Ars Math. Contemp..

[34]  Ernesto Estrada.  Protein bipartivity and essentiality in the yeast protein-protein interaction network. , 2006, Journal of proteome research.

[35]  Milan Randic,et al.  Distance/Distance Matrixes , 1994, J. Chem. Inf. Comput. Sci..

[36]  Ernesto Estrada,et al.  Edge Adjacency Relationships and a Novel Topological Index Related to Molecular Volume , 1995, J. Chem. Inf. Comput. Sci..

[37]  Milan Randić.  On structural ordering and branching of acyclic saturated hydrocarbons , 1998 .

[38]  Andrey A. Dobrynin,et al.  The Szeged Index and an Analogy with the Wiener Index , 1995, J. Chem. Inf. Comput. Sci..

[39]  Sandi Klavzar,et al.  An Algorithm for the Calculation of the Szeged Index of Benzenoid Hydrocarbons , 1995, J. Chem. Inf. Comput. Sci..

[40]  Douglas J. Klein,et al.  Resistance-Distance Sum Rules* , 2002 .

[41]  Frank Harary,et al.  Graph Theory , 2016 .

[42]  Catherine A. Gorini.  Geometry at work : a collection of papers showing applications of geometry , 2000 .

[43]  J. Platt.  Prediction of Isomeric Differences in Paraffin Properties , 1952 .

[44]  Xiaofeng Guo,et al.  Wiener matrix: Source of novel graph invariants , 1993, J. Chem. Inf. Comput. Sci..

[45]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[46]  István Lukovits,et al.  On the Definition of the Hyper-Wiener Index for Cycle-Containing Structures , 1995, J. Chem. Inf. Comput. Sci..