On Eccentric Connectivity Index and Polynomial of Thorn Graph

Nilanjan De

Department of Basic Science, Humanities and Social Science (Mathematics), Calcutta Institute of Engineering and Management, Kolkata, India.

DOI: 10.4236/am.2012.38139                

Abstract

The eccentric connectivity index based on degree and eccentricity of the vertices of a graph is a widely used graph invariant in mathematics. In this paper we present the explicit generalized expressions for the eccentric connectivity index and polynomial of the thorn graphs, and then consider some particular cases.

Keywords

Ecentricity; Eccentric Connectivity Index; Eccentric Connectivity Polynomial; Thorn Graphs

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	V. Sharma, R. Goswami and A. K. Madan, “Eccentric Connectivity Index: A Novel Highly Discriminating Topological Descriptor for Structure-Property and Structure-Activity Studies,” Journal of Chemical Information and Modeling, Vol. 37, No. 2, 1997, pp. 273-282. doi:10.1021/ci960049h
[2]	I. Gutman, “Distance in Thorny Graph,” Publications de l’Institut Mathématique (Beograd), Vol. 63, 1998, pp. 31-36.
[3]	A. Ili? and I. Gutman, “Eccentric Connectivity Index of Chemical Trees,” MATCH—Communications in Mathematical and in Computer Chemistry, Vol. 65, 2011, pp. 731-744.
[4]	B. Zhou and Z. Du, “On Eccentric Connectivity Index,” MATCH—Communications in Mathematical and in Computer Chemistry, Vol. 63, 2010, pp. 181-198.
[5]	P. Dankelmann, W. Goddard and C. S. Swart, “The Average Eccentricity of a Graph and Its Subgraphs,” Utilitas Mathematica, Vol. 65, 2004, pp. 41-51.
[6]	D. Vuki?evi? and A. Graovac, “Note on the Comparison of the First and Second Normalized Zagreb Eccentricity Indices,” Acta Chimica Slovenica, Vol. 57, 2010, pp. 524-528.
[7]	T. Do?li?, M. Saheli and D. Vuki?evi?, “Eccentric Connectivity Index: Extremal Graphs and Values,” Iranian Journal of Mathematical Chemistry, Vol. 1, No. 2, 2010, pp. 45-56.
[8]	K. C. Das and N. Trinajsti?, “Relationship between the Eccentric Connectivity Index and Zagreb Indices,” Computers & Mathematics with Applications, Vol. 62, No. 4, 2011, pp. 1758-1764. doi:10.1016/j.camwa.2011.06.017
[9]	L. Zhang and H. Hua, “The Eccentric Connectivity Index of Unicyclic Graphs,” International Journal of Contemporary Mathematical, Vol. 5, No. 46, 2010, pp. 2257-2262.
[10]	J. Yang and F. Xia, “The Eccentric Connectivity Index of Dendrimers,” International Journal of Contemporary Mathematical, Vol. 5, No. 45, 2010, pp. 2231-2236.
[11]	M. Ghorbani and M. Hemmasi, “Eccentric Connectivity Polynomial of C12n+4 Fullerenes,” Digest Journal of Nanomaterials and Biostructures, Vol. 4, No. 3, 2009, pp. 545-547.
[12]	B. Zhou and D. Vuki?evi?, “On Wiener-Type Polynomials of Thorn Graphs,” Journal of Chemometrics, Vol. 23, No. 12, 2009, pp. 600-604.
[13]	D. Bonchev and D. J. Klein, “On the Wiener Number of Thorn Trees, Stars, Rings, and Rods,” Croatica Chemica Acta, Vol. 75, No. 2, 2002, pp. 613-620.
[14]	A. Heydari and I. Gutman, “On the Terminal Wiener Index of Thorn Graphs,” Kragujevac Journal of Science, Vol. 32, 2010, pp. 57-64.
[15]	B. Zhou, “On Modified Wiener Indices of Thorn Trees,” Kragujevac Journal of Mathematics, Vol. 27, 2005, pp. 5-9.
[16]	D. Vuki?evi?, B. Zhou and N. Trinajsti?, “Altered Wiener Indices of Thorn Trees,” Croatica Chemica Acta, Vol. 80, No. 2, 2007, pp. 283-285.
[17]	H. B. Walikar, H. S. Ramane, L. Sindagi, S. S. Shirakol and I. Gutman, “Hosoya Polynomial of Thorn Trees, Rods, Rings, and Stars,” Kragujevac Journal of Science, Vol. 28, 2006, pp. 47-56.
[18]	S. Li, “Zagreb Polynomials of Thorn Graphs,” Kragujevac Journal of Science, Vol. 33, 2011, pp. 33-38.