Table of Contents
International Journal of Computational Mathematics
Volume 2014, Article ID 436140, 8 pages
http://dx.doi.org/10.1155/2014/436140
Research Article

Modified Eccentric Connectivity of Generalized Thorn Graphs

1Department of Basic Sciences and Humanities (Mathematics), Calcutta Institute of Engineering and Management, Kolkata 700 040, India
2Department of Mathematics, National Institute of Technology, Durgapur 713 209, India
3Department of Mathematics, Aliah University, DN 20, Sector V, Salt Lake, Kolkata 700 091, India

Received 26 June 2014; Accepted 9 December 2014; Published 22 December 2014

Academic Editor: Sheung-Hung Poon

Copyright © 2014 Nilanjan De et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The thorn graph of a given graph is obtained by attaching pendent vertices to each vertex of . The pendent edges, called thorns of , can be treated as or , so that a thorn graph is generalized by replacing by and by and the respective generalizations are denoted by and . The modified eccentric connectivity index of a graph is defined as the sum of the products of eccentricity with the total degree of neighboring vertices, over all vertices of the graph in a hydrogen suppressed molecular structure. In this paper, we give the modified eccentric connectivity index and the concerned polynomial for the thorn graph and the generalized thorn graphs and .

1. Introduction

Let be a simple connected graph with vertex set and edge set , so that and . Let the vertices of be labeled as . For any vertex the number of neighbors of is defined as the degree of the vertex and is denoted by . Let denote the set of vertices which are the neighbors of the vertex , so that . Also let , that is, sum of degrees of the neighboring vertices of . The distance between the vertices and is equal to the length of the shortest path connecting and . Also for a given vertex , the eccentricity is the largest distance from to any other vertices of and the sum of eccentricities of all the vertices of is denoted by [1]. The eccentric connectivity index of a graph was proposed by Sharma et al. [2]. A lot of results related to chemical and mathematical study on eccentric connectivity index have taken place in the literature [35]. There are numerous modifications of eccentric connectivity index reported in the literature till date. These include edge versions of eccentric connectivity index [6], eccentric connectivity topochemical index [7], augmented eccentric connectivity index [8], superaugmented eccentric connectivity index [9], and connective eccentricity index [10]. A modified version of eccentric connectivity index was proposed by Ashrafi and Ghorbani [11].

Similar to other topological polynomials, the corresponding polynomial, that is, the modified eccentric connectivity polynomial of a graph, is defined as so that the modified eccentric connectivity index is the first derivative of this polynomial for . Several studies on this modified eccentric connectivity index are also found in the literature. In [11], the modified eccentric connectivity polynomials for three infinite classes of fullerenes were computed. In [12], a numerical method for computing modified eccentric connectivity polynomial and modified eccentric connectivity index of one-pentagonal carbon nanocones was presented. In [13], some exact formulas for the modified eccentric connectivity polynomial of Cartesian product, symmetric difference, disjunction, and join of graphs were presented. Some upper and lower bounds for this modified eccentric connectivity index are recently obtained by the present authors [14].

The first and the second Zagreb indices of , denoted by and , respectively, are two of the oldest topological indices introduced in [15] by Gutman and Trinajstić and were defined as

Let be a -tuple of nonnegative integers. The thorn graph is the graph obtained from by attaching pendent vertices to the vertex of . In this paper, we assume . These pendent vertices are termed as thorns. The concept of thorn graphs was first introduced by Gutman [16]. A lot of studies on thorn graphs for different topological indices are made by several researchers in the recent past [1724]. Very recently, De [25, 26] studied two eccentricity related topological indices, such as eccentric connectivity index and augmented eccentric connectivity indices, on thorn graphs.

The thorns of the thorn graph can be treated as or , so that the thorn graph can be generalized by replacing by and by and the generalizations are, respectively, denoted by and . In the following section, we present the explicit expressions of the modified eccentric connectivity index of thorn graph and its generalized forms and .

2. Evaluation of Modified Eccentric Connectivity Index

The eccentric connectivity index [2] and connective eccentric index [10] of a graph are defined as

The modified eccentric connectivity index [11] is defined as

Total eccentricity index is defined as . Total eccentricity index of the generalized hierarchical product of graphs has been studied by De et al. recently [14].

Since the modified eccentric connectivity index is likely to have an application in drug discovery process, therefore, we evaluate the index in comparison to some other well known indices in this section.

The two graphs shown in Figure 1 have the same eccentricity connectivity index and connective eccentricity index, but they have different modified eccentric connectivity indices.

Figure 1: Two graphs with the same and , but with different .

To evaluate the modified eccentric connectivity index (MECI) in terms of degeneracy and intercorrelation with other well known indices we compute different topological indices such as eccentric connectivity index (ECI), total eccentricity index (TEI), connective eccentricity index, (CEI) and augmented eccentric connectivity index (AECI) for octane isomers as given in Table 1.

Table 1: Different topological indices of octane isomers.

For modified eccentric connectivity index we observe that maximum value = 136, minimum value = 93, ratio = 1.46, and degeneracy = 4/18.

Intercorrelation of modified eccentric connectivity index with some well known vertex eccentricity based topological indices is given in Table 2.

Table 2: Intercorrelation of indices.

3. Main Results

First we find the modified eccentric connectivity index of the thorn graph in terms of modified eccentric connectivity index of , total eccentricity of , and first Zagreb index of .

Theorem 1. For any simple connected graph , and are related as , where and is the thorn graph of .

Proof. Let and , where . Here, are the set of degree one vertices attached to the vertices in and . Let the vertices of the set be denoted by for .
Then the degree of the vertices in is given by , for . Hence, for , and . Similarly the eccentricity of the vertices , in is given by , for , and the eccentricity of the vertices is given by , for and .
Then the modified eccentric connectivity index of is given by Now,
Combining the above equations, we get

The eccentric connectivity polynomial and total eccentricity polynomial of are defined as and , respectively. It is easy to see that the eccentric connectivity index and the total eccentricity of a graph can be obtained from the corresponding polynomials by evaluating their first derivatives at .

Now we express the modified eccentric polynomial of a thorn graph in terms of eccentric connectivity polynomial, modified eccentric connectivity polynomial, and total eccentric polynomial of the parent graph.

Theorem 2. For any simple connected graph , the polynomials and are related as , where is the thorn graph of .

Proof. Following the previous theorem, the modified eccentric connectivity polynomial of is given by .
Now,
After addition, we get

It can be easily verified that expression (7) is obtained by differentiating (9) with respect to and by putting .

Let be the graph obtained from by attaching complete graphs of order , that is, , at every vertex of . Let the vertices attached to the vertex be denoted by , ; . Let the vertex be identified with , ; .

Theorem 3. For any simple connected graph , and are related as , where is the graph obtained from by attaching complete graphs at each vertex of .

Proof. The eccentricities of the vertices of are given by , for , and , for ; ; .
The degree of the vertices of is given by , for , and , for ; ; .
Thus, + , for , and , for ; ; .
Therefore the modified eccentric connectivity index of is given by
Now,
Adding, we get

Since for , the generalized thorn graph reduces to the usual thorn graph , Theorem 1 follows from Theorem 3 by substituting .

In the following, we find the modified eccentric connectivity polynomial of the graph in terms of eccentric connectivity polynomial, modified eccentric connectivity polynomial, and total eccentric polynomial of the parent graph .

Theorem 4. For any simple connected graph , the is given by , where is the graph obtained from by attaching complete graphs at each vertex of .

Proof. From definition, the modified eccentric connectivity polynomial of is given by
Now,
Similarly,
Adding the above two, we get

It can also be verified that expression (11) is obtained by differentiating (15) with respect to and by putting . Also Theorem 2 follows from Theorem 4 by substituting .

Let us now construct a graph by attaching paths of order at each vertex of . The vertices of the th path attached to are denoted by ; . Let the vertex be identified with the th vertex of . Clearly the resulting graph consists of number of vertices.

Theorem 5. For any simple connected graph with vertices, and are related as , where is the graph obtained from by attaching paths each of length at each vertex of .

Proof. The eccentricities of the vertices of are given by , for , and , for ; ; .
The degrees of the vertices of are given by , for ; , for ; ; and , for ; .
Thus, , for ; , for ; ; ; , for ; ; , for ; and , for ; .
Therefore, the modified eccentric connectivity index of is given by
Now,
Also,
Combining the above, we have

Since the generalized thorn graph also reduces to the usual thorn graph for , Theorem 1 follows from Theorem 5 by substituting .

In the following, we find the modified eccentric connectivity polynomial of the graph in terms of eccentric connectivity polynomial, modified eccentric connectivity polynomial, and total eccentric polynomial of the parent graph .

Theorem 6. For any simple connected graph , the modified eccentric connectivity polynomial is given by + , where is the graph obtained from by attaching paths each of length at each vertex of .

Proof. The modified eccentric connectivity polynomial of is given by
Now, proceeding as above theorem, we have
Again,
Combining the above two, we get

Here also, differentiating (23) with respect to and putting , we get relation (19).

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The authors are grateful to anonymous reviewer whose suggestions improved the presentation of the paper.

References

  1. P. Dankelmann, W. Goddard, and C. S. Swart, “The average eccentricity of a graph and its subgraphs,” Utilitas Mathematica, vol. 65, pp. 41–51, 2004. View at Google Scholar · View at MathSciNet · View at Scopus
  2. 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 Computer Sciences, vol. 37, no. 2, pp. 273–282, 1997. View at Google Scholar · View at Scopus
  3. T. Došlic, M. Saheli, and D. Vukičević, “Eccentric connectivity index: extremal graphs and values,” Iranian Journal of Mathematical Chemistry, vol. 1, pp. 45–56, 2010. View at Google Scholar
  4. A. Ilić and I. Gutman, “Eccentric connectivity index of chemical trees,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 65, pp. 731–744, 2011. View at Google Scholar
  5. B. Zhou and Z. Du, “On eccentric connectivity index,” Match Communications in Mathematical and in Computer Chemistry, vol. 63, no. 1, pp. 181–198, 2010. View at Google Scholar · View at MathSciNet
  6. X. Xu and Y. Guo, “The edge version of eccentric connectivity index,” International Mathematical Forum, vol. 7, no. 6, pp. 273–280, 2012. View at Google Scholar · View at MathSciNet
  7. V. Kumar, S. Sardana, and A. K. Madan, “Predicting anti-HIV activity of 2,3-diaryl-1,3-thiazolidin-4-ones: computational approach using reformed eccentric connectivity index,” Journal of Molecular Modeling, vol. 10, no. 5-6, pp. 399–407, 2004. View at Publisher · View at Google Scholar · View at Scopus
  8. T. Došlić and M. Saheli, “Augmented eccentric connectivity index,” Miskolc Mathematical Notes, vol. 12, no. 2, pp. 149–157, 2011. View at Google Scholar
  9. H. Dureja and A. K. Madan, “Superaugmented eccentric connectivity indices: new-generation highly discriminating topological descriptors for QSAR/QSPR modeling,” Medicinal Chemistry Research, vol. 16, no. 7–9, pp. 331–341, 2007. View at Publisher · View at Google Scholar · View at Scopus
  10. S. Gupta, M. Singh, and A. K. Madan, “Connective eccentricity index: a novel topological descriptor for predicting biological activity,” Journal of Molecular Graphics and Modelling, vol. 18, no. 1, pp. 18–25, 2000. View at Publisher · View at Google Scholar · View at Scopus
  11. A. R. Ashrafi and M. Ghorbani, “A study of fullerenes by MEC polynomials,” Electronic Materials Letters, vol. 6, no. 2, pp. 87–90, 2010. View at Publisher · View at Google Scholar · View at Scopus
  12. M. Alaeiyan, J. Asadpour, and R. Mojarad, “A numerical method for MEC polynomial and MEC index of one-pentagonal carbon nanocones,” Fullerenes, Nanotubes and Carbon Nanostructures, vol. 21, no. 10, pp. 825–835, 2013. View at Publisher · View at Google Scholar · View at Scopus
  13. A. R. Ashrafi, M. Ghorbani, and M. A. Hossein-Zadeh, “The eccentric connectivity polynomial of some graph operations,” Serdica Journal of Computing, vol. 5, no. 2, pp. 101–116, 2011. View at Google Scholar · View at MathSciNet
  14. N. De, S. M. A. Nayeem, and A. Pal, “Total eccentricity index of the generalized hierarchical product of graphs,” International Journal of Applied and Computational Mathematics, 2014. View at Publisher · View at Google Scholar
  15. I. Gutman and N. Trinajstić, “Graph theory and molecular orbitals. Total π-electron energy of alternant hydrocarbons,” Chemical Physics Letters, vol. 17, no. 4, pp. 535–538, 1972. View at Publisher · View at Google Scholar · View at Scopus
  16. I. Gutman, “Distance of thorny graphs,” Publications de l'Institut Mathématique, vol. 63, pp. 31–36, 1998. View at Google Scholar · View at MathSciNet
  17. D. Bonchev and D. J. Klein, “On the wiener number of thorn trees, stars, rings, and rods,” Croatica Chemica Acta, vol. 75, no. 2, pp. 613–620, 2002. View at Google Scholar · View at Scopus
  18. A. Heydari and I. Gutman, “On the terminal Wiener index of thorn graphs,” Kragujevac Journal of Science, vol. 32, pp. 57–64, 2010. View at Google Scholar
  19. K. M. Kathiresan and C. Parameswaran, “Certain generalized thorn graphs and their Wiener indices,” Journal of Applied Mathematics and Informatics, vol. 30, no. 5-6, pp. 793–807, 2012. View at Google Scholar
  20. S. Li, “Zagreb polynomials of thorn graphs,” Kragujevac Journal of Science, vol. 33, pp. 33–38, 2011. View at Google Scholar
  21. D. Vukičević, B. Zhou, and N. Trinajstićc, “Altered Wiener indices of thorn trees,” Croatica Chemica Acta, vol. 80, no. 2, pp. 283–285, 2007. View at Google Scholar · View at Scopus
  22. 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, pp. 47–56, 2006. View at Google Scholar
  23. B. Zhou, “On modified Wiener indices of thorn trees,” Kragujevac Journal of Mathematics, vol. 27, pp. 5–9, 2005. View at Google Scholar · View at MathSciNet
  24. B. Zhou and D. Vukičević, “On Wiener-type polynomials of thorn graphs,” Journal of Chemometrics, vol. 23, no. 12, pp. 600–604, 2009. View at Publisher · View at Google Scholar · View at Scopus
  25. N. De, “On eccentric connectivity index and polynomial of thorn graph,” Applied Mathematics, vol. 3, pp. 931–934, 2012. View at Google Scholar
  26. N. De, “Augmented eccentric connectivity index of some thorn graphs,” International Journal of Applied Mathematical Research, vol. 1, no. 4, pp. 671–680, 2012. View at Google Scholar