Abstract
The eccentric connectivity index \(\xi ^c(G)\) of a connected graph G is defined as \(\xi ^c(G) =\sum _{v \in V(G)}{deg(v) e(v)},\) where deg(v) is the degree of vertex v and e(v) is the eccentricity of v. The eccentric graph, \(G_e\), of a graph G has the same set of vertices as G, with two vertices u, v adjacent in \(G_e\) if and only if either u is an eccentric vertex of v or v is an eccentric vertex of u. In this paper, we obtain a formula for the eccentric connectivity index of the eccentric graph of a regular dendrimer. We also derive a formula for the eccentric connectivity index for the second iteration of eccentric graph of regular dendrimer.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bonchev, D., Rouvray, D.H. (eds.): Chemical Graph Theory: Introduction and Fundamentals. Taylor and Francis, Milton Park (1991)
Newkome, G.R., Moorefield, C.N., Vogtle, F.: Dendrimers and Dendrons: Concepts, Syntheses, Applications. Wiley-VCH, Verlag GmbH and Co.KGaA, Weinheim (2002)
Weiner, H.J.: Structural determination of Paraffin boiling points. J. Am. Chem. Soc 69, 17–20 (1947)
Stevanovic, D.: Comparing the Zagreb indices of the NEPS of graphs. Appl. Math. Comput. 219(3), 1082–1086 (2012)
Das, K.C., Gutman, I., Furtula, B.: On atom-bond connectivity index. Filomat 26(4), 733–738 (2012)
Sharma, V., Goswami, R., Madan, A.K.: Eccentric connectivity index: a novel highly discriminating topological descriptor for structural property and structure activity studies. J. Chem. Inform. Model. 37, 273–282 (1997)
Ahmadi, M.B., Sadeghimehr, M.: Second-order connectivity index of an infinite class of dendrimer nanostars. Digest J. Nanomater. Biostruct. 4, 639–643 (2009)
Ghorbani, M., Jalali, M.: A simple algorithm for computing topological indices of dendrimers. Iran. J. Math. Sci. Inform. 2, 17–23 (2007)
Ghorbani, M., Jalali, M.: Some topological indicies of nanostar dendrimers. Iran. J. Math. Chem. 1, 57–65 (2010)
Shabani, H.: Computing vertex PI index of tetrathiafulvalene dendrimers. Iran. J. Math. Chem. 1, 125–130 (2010)
Yang, J., Xia, F.: The eccentric connectivity index of Dendrimers. Int. J. Contem. Math. Sci. 5, 2231–2236 (2010)
Alikhani, S., Iranmanesh, M.A.: Eccentric connectivity polynomials of an infinite family of dendrimer. Digest J. Nanomater. Biostruct. 6(1), 253–257 (2011)
Buckley, F., Harary, F.: Distance in Graphs. Addison-Wesley, Redwood City (1990)
Akiyama, J., Ando, K., Avis, D.: Eccentric graphs. Discrete Math. 56, 1–6 (1985)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nagar, A.K., Sriram, S. On Eccentric Connectivity Index of Eccentric Graph of Regular Dendrimer. Math.Comput.Sci. 10, 229–237 (2016). https://doi.org/10.1007/s11786-016-0259-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11786-016-0259-z