Table of Contents Author Guidelines Submit a Manuscript
Journal of Applied Mathematics
Volume 2014, Article ID 241712, 8 pages
http://dx.doi.org/10.1155/2014/241712
Research Article

Zagreb Eccentricity Indices of the Generalized Hierarchical Product Graphs and Their Applications

1School of Mathematics, Shandong University, Jinan 250100, China
2Department of Mathematics, Changji University, Changji 831100, China

Received 12 October 2013; Accepted 22 December 2013; Published 10 February 2014

Academic Editor: Juan Manuel Peña

Copyright © 2014 Zhaoyang Luo and Jianliang Wu. 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

Let be a connected graph. The first and second Zagreb eccentricity indices of are defined as and , where is the eccentricity of the vertex in and . Suppose that is the generalized hierarchical product of two connected graphs and . In this paper, the Zagreb eccentricity indices and of are computed. Moreover, we present explicit formulas for the and of S-sum graph, Cartesian, cluster, and corona product graphs by means of some invariants of the factors.

1. Introduction

A topological index is a real number associated with chemical constitution purporting for correlation of chemical structure with various physical properties, chemical reactivity, or biological activity, which is used to understand properties of chemical compounds in theoretical chemistry [1].

Up to now, hundreds of topological indices have been defined in chemical literatures, various applications of these topological indices have been found, and many mathematical properties are also investigated. Wiener index is the first topological index, introduced by American chemist Wiener, for investigating boiling points of alkanes in 1947 [2]. The well known degree-based topological indices are the first and second Zagreb indices and , which have been introduced by Gutman and Trinajstić [3] and applied to study molecular chirality in quantitative structure-activity relationship (QSAR) and quantitative structure-property relationship (QSPR) analysis and so forth. Resently, the first and second Zagreb eccentricity indices and have been introduced by Ghorbani and Hosseinzadeh [4] and Vukičević and Graovac [5] as the revised version of the Zagreb indices and , respectively. They computed the Zagreb eccentricity indices of some composite graphs and showed that holds for all acyclic and unicyclic graphs and that neither this nor the opposite inequality holds for all bicyclic graphs. For further results of the Zagreb eccentricity indices, we encourage the reader to refer to [68].

In 2009, Spain mathematicians Barrière and coauthors [9] introduced a new composite graph, namely, hierarchical product graph. In the same year, this team also reported a generalization of both Cartesian and the hierarchical product of graphs, namely, the generalized hierarchical product of graphs in [10]. After that, many results for some topological indices of the (generalized) hierarchical product of graphs are reported; see [1117].

In this paper, the Zagreb eccentricity indices of the generalized hierarchical product graph are computed and as some special cases of , the Zagreb eccentricity indices of the Cartesian product graph , the -sum graph , and the cluster product graph are determined, respectively. Moreover, as applications, we present explicit formulas for the and indices of the nanotorus , the nanotubes , the zig-zay polyhex nanotube , the hexagonal chain , and so forth.

2. Preliminaries

Throughout this paper, all graphs are simple, finite, and undirected. For terminology and notations that are not defined here, we refer the reader to West [18].

Let be a graph with the vertex set , the edge set , and an incidence function that associates with each edge of , an unordered pair of vertices of . If is an edge and and are vertices such that , then is said to join and , and the vertices and are called the ends of . The cardinality of and is denoted by and , respectively. We denote the degree and the neighborhood of a vertex of by and ; then . As usual, the distance between vertices and of a connected graph , denoted by , is defined as the number of edges in a shortest path connecting the vertices and . Suppose that and . The eccentricity of a vertex in is the largest distance between and any other vertex of ; that is, . For two graphs and , if there exist two bijections and such that if and only if , then we say that and are isomorphic, denoted by . Let denote a certain topological index of . In general, if , then .

The total eccentricity and the eccentric connectivity indices and of graph are defined as and , respectively. The Zagreb indices of are defined as and . Very recently, the topological indices based on vertex eccentricities attracted some attention in chemistry. In an analogy with the Zagreb indices, the first and second Zagreb eccentricity indices and of a connected graph are defined by [4, 5]. That is,

Lemma 1 (see [4]). Let be the complete graph of order ; then , , , and .

Lemma 2 (see [4]). Let be the cycle of length ; then , , and .

Lemma 3 (see [4]). Let be the path on vertices. Then

3. Zagreb Eccentricity Indices of Generalized Hierarchical Product Graphs

In this section, we calculate the Zagreb eccentricity indices of the generalized hierarchical product graphs.

Definition 4 (see [9]). Let and be two connected graphs; . Then the generalized hierarchical product is the graph with vertex set and vertices and are adjacent if and only if and or and .

Given a connected graph and , a path connecting vertices and through is a -path of containing some vertex (vertex could be the vertex or vertex ). Then the distance through between and is denoted by , which is the length of the shortest path in . Note that if one of the vertex and belongs to , then ; see [13]. Similarly, we define some invariants related to in as follows:, , ; see [15],; see [15],, , .

Theorem 5 (see [15]). Let graphs and be connected; . Then

Lemma 6 (see [10]). Let and be two connected graphs and . Then

Theorem 7. Let graphs and be connected; . Then
(i)
(ii)

Proof. Let and .
(i) By the definition of Zagreb eccentricity index and Lemma 6, we have
(ii) We partition the edges of into two subsets and , as follows: From the definition of Zagreb eccentricity index and Lemma 6, we get This completes the proofs.

Definition 8 (see [13]). Let and be two connected graphs. Then the Cartesian product has the vertex set and vertices and are adjacent if and only if and or and , where and .

Note that if , then . So by Theorem 7, the following corollary is obvious.

Corollary 9. Let and be two connected graphs. Then

Remark 10. Equation (14) corrects the corresponding Corollary  9 in [4]. According to (14), we recompute the second Zagreb eccentricity indices of the nanotubes as below (see Example 11).

Example 11. Using the results as above, it is easy to obtain the first and second Zagreb eccentricity indices of the nanotorus and the nanotubes and . By Corollary 9 and Lemmas 2 and 3, we have

Lemma 12 (see [19]). Let be Cartesian product of connected graphs . Then(a).(b).(c).

Corollary 13. Let be Cartesian product of simple connected graphs . Then

Proof. The case is proved in Corollary 9. We prove the assertion by induction. Suppose the result is valid for graphs. Then by Lemma 12, we have This completes the proof.

Example 14. The Hamming graph . Thus, by Corollary 13 and Lemma 1, we have . For , we attain the -dimensional hypercubes . Therefore, .

Remark 15. Ghorbani and Hosseinzadeh computed the second Zagreb eccentricity index of in [4]. Here, we can also obtain the explicit formula of by induction.
For two connected graphs , we note that if ; then by Theorem 5, we have
By induction, we can easily prove that
Therefore, by Corollary 9, Lemma 12 and the formula as above, using a similar method of proof in Corollary 13, we can obtain Corollary 16.

Corollary 16 (see [4]). Let be Cartesian product of graphs . Then

From Corollaries 13 and 16, the following corollary is obvious.

Corollary 17. Let be Cartesian product of connected graphs . Then

4. Zagreb Eccentricity Indices of -Sum Graphs

Let be a connected graph. The vertices of a Line graph   are the edges of . Two edges of that share a vertex are considered to be adjacent in . A Subdivision graph   is the graph obtained by inserting an additional vertex in each edge of . That is, each edge of is replaced by a path of length two.

Definition 18 (see [15]). For two connected graphs and , the -sum of and is a graph with vertex set and vertices and are adjacent if and only if and or and .

Note that if , then . So by Theorem 7, we can compute the Zagreb eccentricity indices of easily.

Lemma 19 (see [15]). Let and be two connected graphs. If , then we have(a), ,(b)for each vertex , we have ,(c)for each vertex , we have .

Lemma 20 (see [15]). Let () be a tree with vertices. If , then(a)for each vertex , we have ,(b)for each vertex , we have .

Theorem 21. Let be a tree with order () and let be a connected graph; . Then
(i)
(ii) where .

Proof. (i) We start to calculate . By Lemmas 19 and 20, we have Combing these with (8) in Theorem 7, we can obtain the corresponding result.
(ii) Now, let us compute . By Lemmas 19 and 20, we get Combing these results with (9) in Theorem 7, we obtain the desired result.

Clearly, if is even, then . Otherwise, if is odd, then . So, the following lemma holds.

Lemma 22. Let be a path of order . Then

Example 23. Suppose is a linear hexagonal chain with hexagons (see Figure 1); note that . Thus, by Lemmas 3 and 22 and Theorem 21, we have

241712.fig.001
Figure 1: The linear hexagonal chain with hexagons .

Let be an integer with and . Note that for any vertex . Then by Theorem 7, we can obtain the following theorem.

Theorem 24. Let   () be a cycle and let be an arbitrary connected graph; . Then

Example 25. Let be an integer with and let be the zigzag polyhex nanotube (see Figure 2); then . By Theorem 24 and Lemma 3, we have and .

241712.fig.002
Figure 2: The zig-zay polyhex nanotube .

5. Zagreb Eccentricity Indices of Cluster and Corona Product Graphs

The cluster product, corona product, and join of two graphs are important graph operations defined as below.

Definition 26 (see [20]). The cluster product graph is obtained by taking one copy of and copies of a rooted graph and by identifying the root of the th copy of with the th vertex of , .

Definition 27 (see [20]). The corona product graph is obtained by taking one copy of and copies of and by joining each vertex of the th copy of to the th vertex of , .

Definition 28 (see [20]). The join graph : ; .

Let and be connected graphs; is a root-vertex of . Note that; if , then . We define and .

Theorem 29. Let and be two connected graphs; is a root-vertex of . Then

Proof. Let . Then On the other hand, . Note that . Thus, combing these results with Theorem 5, we can obtain (29).
Similarly, we can determine (30) and (31) in terms of Theorem 7, respectively.

Let and be two simple graphs. If and , then we say that is an -graph. According to the definitions of the cluster and corona products, if is an -graph and is a -graph, then is an -graph and is an -graph.

Corollary 30. Let be a connected -graph and is an arbitrary -graph. Then

Proof. For any -graph , let be the root-vertex of graph (the join of graphs and , is the unique vertex in ). Then is a -graph. It is easy to see that , . Note that ; hence . Equation (33) is obtained by (29). Moreover, we note that and . Hence, by (30) and (31), using the same method as above, the corresponding equations (34) and (35) are also obtained, respectively.

As applications, we present some examples as below, these results can be attained by means of Corollary 30, Lemmas 2, and 3.

Example 31. The following equations hold:

Example 32. Let be an arbitrary graph with vertices. Then

Example 33. The following equations hold:

Conflict of Interests

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

Acknowledgments

The authors would like to thank the referee for many helpful comments and suggestions, which have greatly improved the presentation of the paper. This work is supported by the National Natural Science Foundation of China (no. 11271006), the Scientific Research Programs of the Higher Education Institution of Xinjiang (XJEDU2012I37,I38), and the Science Foundation of Changji University (2012YJYB003).

References

  1. I. Gutman and O. E. Polansky, Mathematical Concepts in Organic Chemistry, Springer, Berlin, Germany, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  2. H. Wiener, “Structural determination of paraffin boiling points,” Journal of the American Chemical Society, vol. 69, no. 1, pp. 17–20, 1947. View at Google Scholar · View at Scopus
  3. I. Gutman and N. Trinajstić, “Graph theory and molecular orbitals. Total φ-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
  4. M. Ghorbani and M. A. Hosseinzadeh, “A new version of Zagreb indices,” Filomat, vol. 26, no. 1, pp. 93–100, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. D. Vukičević and A. Graovac, “Note on the comparison of the first and second normalized Zagreb eccentricity indices,” Acta Chimica Slovenica, vol. 57, no. 3, pp. 524–528, 2010. View at Google Scholar
  6. K. Ch. Das, D.-W. Lee, and A. Graovac, “Some properties of the Zagreb eccentricity indices,” Ars Mathematica Contemporanea, vol. 6, no. 1, pp. 117–125, 2013. View at Google Scholar · View at MathSciNet
  7. Z. Du, B. Zhou, and N. Trinajstić, “Extremal properties of the Zagreb eccentricity indices,” Croatica Chemica Acta, vol. 85, no. 3, pp. 359–362, 2012. View at Google Scholar
  8. R. Xing, B. Zhou, and N. Trinajstić, “On zagreb eccentricity indices,” Croatica Chemica Acta, vol. 84, no. 4, pp. 493–497, 2011. View at Publisher · View at Google Scholar · View at Scopus
  9. L. Barrière, F. Comellas, C. Dalfó, and M. A. Fiol, “The hierarchical product of graphs,” Discrete Applied Mathematics, vol. 157, no. 1, pp. 36–48, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. L. Barrière, C. Dalfó, M. A. Fiol, and M. Mitjana, “The generalized hierarchical product of graphs,” Discrete Mathematics, vol. 309, no. 12, pp. 3871–3881, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. Arezoomand and B. Taeri, “Applications of generalized hierarchical product of graphs in computing the Szeged index of chemical graphs,” MATCH. Communications in Mathematical and in Computer, vol. 64, no. 3, pp. 591–602, 2010. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. Arezoomand and B. Taeri, “Zagreb indices of the generalized hierarchical product of graphs,” MATCH. Communications in Mathematical and in Computer, vol. 69, no. 1, pp. 131–140, 2013. View at Google Scholar · View at MathSciNet
  13. M. Eliasi and A. Iranmanesh, “The hyper-Wiener index of the generalized hierarchical product of graphs,” Discrete Applied Mathematics, vol. 159, no. 8, pp. 866–871, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Eliasi and A. Iranmanesh, “Hosoya polynomial of hierarchical product of graphs,” MATCH. Communications in Mathematical and in Computer, vol. 69, no. 1, pp. 111–119, 2013. View at Google Scholar · View at MathSciNet
  15. B. Eskender and E. Vumar, “Eccentric connectivity index and eccentric distance sum of some graph operations,” Transactions on Combinatorics, vol. 2, no. 1, pp. 103–111, 2013. View at Google Scholar
  16. K. Pattabiraman and P. Paulraja, “Vertex and edge Padmakar-Ivan indices of the generalized hierarchical product of graphs,” Discrete Applied Mathematics, vol. 160, no. 9, pp. 1376–1384, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. M. Tavakoli, F. Rahbarnia, and A. R. Ashrafi, “Further results on hierarchical product of graphs,” Discrete Applied Mathematics, vol. 161, no. 7-8, pp. 1162–1167, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. D. B. West, Introduction to Graph Theory, Prentice Hall, Upper Saddle River, NJ, USA, 2nd edition, 1996. View at MathSciNet
  19. M. H. Khalifeh, H. Yousefi-Azari, and A. R. Ashrafi, “Vertex and edge PI indices of Cartesian product graphs,” Discrete Applied Mathematics, vol. 156, no. 10, pp. 1780–1789, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. Y. N. Yeh and I. Gutman, “On the sum of all distances in composite graphs,” Discrete Mathematics, vol. 135, no. 1–3, pp. 359–365, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet