The Eccentric Connectivity Polynomial of some Graph Operations

The eccentric connectivity index of a graph G, � C , was proposed by Sharma, Goswami and Madan. It is defined asC (G) = P u2V (G) degG(u)"G(u), where degG(u) denotes the degree of the vertex x in G and "G(u) = Max{d(u,x) | x ∈ V (G)}. The eccentric connectivity polynomial is a polynomial version of this topological index. In this pa- per, exact formulas for the eccentric connectivity polynomial of Cartesian product, symmetric difference, disjunction and join of graphs are presented. 1. Introduction. Throughout this paper all graphs are assumed to be simple, finite and connected. A function Top from the class of connected graphs into real numbers with the property that Top(G) = Top(H) whenever G and H are isomorphic is known as a topological index in the chemical literature; see (11). There are many examples of such functions, especially those based on distances, which are applicable in chemistry. The Wiener index (19), defined as

[1]  A. K. Madan,et al.  Application of Graph Theory: Relationship of Eccentric Connectivity Index and Wiener's Index with Anti-inflammatory Activity , 2002 .

[2]  Danail Bonchev,et al.  Vertex-weightings for distance moments and thorny graphs , 2007, Discret. Appl. Math..

[3]  A. Ashrafi,et al.  Computing Sadhana polynomial of V-phenylenic nanotubes and nanotori , 2008 .

[4]  W. Imrich,et al.  Product Graphs: Structure and Recognition , 2000 .

[5]  Ali Reza Ashrafi,et al.  The first and second Zagreb indices of some graph operations , 2009, Discret. Appl. Math..

[6]  Vipin Kumar,et al.  Application of Graph Theory: Prediction of Cytosolic Phospholipase A2 Inhibitory Activity of Propan-2-ones , 2006 .

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

[8]  Doslic Tomislav,et al.  Eccentric Connectivity Index: Extremal Graphs and Values , 2010 .

[9]  M. H. Khalifeh,et al.  The Edge Szeged Index of Product Graphs , 2008 .

[10]  I. Gutman,et al.  Eccentric Connectivity Index of Chemical Trees , 2011, 1104.3206.

[11]  Ali Reza Ashrafi,et al.  A matrix method for computing Szeged and vertex PI indices of join and composition of graphs , 2008 .

[12]  I. Gutman,et al.  Graph theory and molecular orbitals. Total φ-electron energy of alternant hydrocarbons , 1972 .

[13]  Yeong-Nan Yeh,et al.  Wiener Polynomials of Some Chemically Interesting Graphs , 2004 .

[14]  ECCENTRIC CONNECTIVITY POLYNOMIAL OF C 12 n + 4 FULLERENES , 2009 .

[15]  Bo Zhou,et al.  ON ECCENTRIC CONNECTIVITY INDEX , 2010 .

[16]  Harish Dureja,et al.  Superaugmented eccentric connectivity indices: new-generation highly discriminating topological descriptors for QSAR/QSPR modeling , 2007, Medicinal Chemistry Research.

[17]  D. West.  Introduction to Graph Theory , 1995 .

[18]  Vipin Kumar,et al.  Predicting anti-HIV activity of 2,3-diaryl-1,3-thiazolidin-4-ones: computational approach using reformed eccentric connectivity index , 2004, Journal of molecular modeling.

[19]  Aleksandar Ilic,et al.  Eccentric connectivity index , 2011, 1103.2515.

[20]  A. K. Madan,et al.  Eccentric Connectivity Index: A Novel Highly Discriminating Topological Descriptor for Structure-Property and Structure-Activity Studies , 1997 .

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

[22]  V. Voloshin.  Introduction to Graph and Hypergraph Theory , 2013 .

[23]  Stephan G. Wagner,et al.  Some new results on distance-based graph invariants , 2009, Eur. J. Comb..

[24]  A. Ashrafi,et al.  The eccentric connectivity index of $TUC_4C_8(R)$ nanotubes , 2011 .

[25]  Viney Lather,et al.  Application of graph theory: Topological models for prediction of CDK-1 inhibitory activity of aloisines , 2005 .

[26]  Ali Reza Ashrafi,et al.  The eccentric connectivity index of nanotubes and nanotori , 2011, J. Comput. Appl. Math..

[27]  Ali Reza Ashrafi,et al.  The PI index of product graphs , 2008, Appl. Math. Lett..

[28]  A. Ashrafi,et al.  The eccentric connectivity index of armchair polyhex nanotubes , 2010 .

[29]  Yeong-Nan Yeh,et al.  The Wiener polynomial of a graph , 1998, math/9801011.