About this Journal Submit a Manuscript Table of Contents
Journal of Applied Mathematics
Volume 2013 (2013), Article ID 681019, 6 pages
http://dx.doi.org/10.1155/2013/681019
Research Article

Bounds on the Distance Energy and the Distance Estrada Index of Strongly Quotient Graphs

1Department of Mathematics, Science Faculty, Selçuk University, 42075 Konya, Turkey
2Department of Studies in Mathematics, University of Mysore, Manasagangotri, Mysore 570 006, India

Received 31 January 2013; Accepted 19 April 2013

Academic Editor: Chong Lin

Copyright © 2013 Ş. Burcu Bozkurt 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 notion of strongly quotient graph (SQG) was introduced by Adiga et al. (2007). In this paper, we obtain some better results for the distance energy and the distance Estrada index of any connected strongly quotient graph (CSQG) as well as some relations between the distance Estrada index and the distance energy. We also present some bounds for the distance energy and the distance Estrada index of CSQG whose diameter does not exceed two. Additionally, we show that our results improve most of the results obtained by Güngör and Bozkurt (2009) and Zaferani (2008).

1. Introduction

Since the distance matrix and related matrices based on graph-theoretical distances are efficient sources of many topological indices that are widely used in theoretical chemistry [1, 2], it is of interest to study spectrum and spectrum-based invariants of these matrices.

Let be a connected graph with vertices and edges and let the vertices of be labeled as . Such a graph will be referred to as connected -graph. Let be the distance matrix of the graph , where denotes the distance (i.e., the length of the shortest path [3]) between the vertices and of . The diameter of the graph , denoted by , is the maximum distance between any two vertices of . The eigenvalues of are said to be the -eigenvalues of . Since is a real symmetric matrix, its eigenvalues are real numbers. So, we can order them so that . For more details on -eigenvalues, especially on , see [312].

The distance energy of the graph is defined as [13] This concept was motivated by the ordinary graph energy which is defined as the sum of absolute values of ordinary graph eigenvalues [1416]. It was also studied intensely in the literature. For instance, Indulal et al. [13] reported lower and upper bounds for the distance energy of graphs whose diameter does not exceed two. In [17] Ramane et al. generalized the results obtained in [13]. Zhou and Ilić [10] established lower bounds for the distance energy of graphs and characterized the extremal graphs. They also discussed upper bounds for the distance energy. Ilić [18] calculated the distance energy of unitary Cayley graphs and presented two families of integral circulant graphs with equal distance energy. Zaferani [19] established an upper bound for the distance energy of strongly quotient graphs. For more results on distance energy, see also the recent papers [6, 20].

Recently, another graph invariant based on graph eigenvalues was put forward in [21]. It was eventually studied under the name Estrada index in [22]. For more details on Estrada index, see [2127]. Motivating the ideas in [21, 22] and considering the distance matrix of the graph , the authors defined the distance Estrada index of as the following [28]: In [28], they also established some lower and upper bounds for this index.

During the past forty years or so enormous amount of research work has been done on graph labeling, where the vertices are assigned values subject to certain conditions. These interesting problems have been motivated by practical problems. Recently, Adiga et al. [29] introduced the notion of strongly quotient graphs and studied these types of graphs. Throughout this paper by a labeling of a graph of order we mean an injective mapping We define the quotient function by if joins and . Note that for any ,  .

A graph with vertices is called a strongly quotient graph if its vertices can be labeled such that the quotient function is injective, that is, the values on the edges are all distinct. For detailed information on graph labeling and strongly quotient graphs, see [19, 23, 29, 30]. Throughout this paper and stand for strongly quotient graph and connected strongly quotient graph of order with maximum number of edges, respectively.

In this paper, we obtain some bounds for the distance energy and the distance Estrada index as well as some relations between and where is . We present some bounds for and of whose diameter does not exceed two. We also show that our results improve most of the results obtained in [19, 28] for .

2. Preliminaries

In this section, we give some lemmas which will be used in our main results.

Lemma 1 (see [31]). Let be nonnegative numbers. Then

Lemma 2 (see [17]). Let be a connected -graph and its -eigenvalues. Then

Lemma 3 (see [13]). Let be a connected -graph and let , where denotes the diameter of the graph . Then

Lemma 4 (see [28]). Let be a connected -graph and be the diameter of . Then The equality holds in (10) if and only if .

Lemma 5 (see [19]). If is a , then is a -eigenvalue of with multiplicity greater than or equal to , where

Lemma 6 (see [19]). If is a , then is a -eigenvalue of with multiplicity greater than or equal to where

3. Bounds on Distance Energy of CSQG

In this section, we will present a better upper bound and a new lower bound for where is with -eigenvalues . Let be the number of positive -eigenvalues of and and are as defined in Lemmas 5 and 6, respectively. For our convenience, we rename the -eigenvalues such that and .

Theorem 7. Let be a connected strongly quotient graph () with vertices and maximum edges . Let and . Then where

Proof. Taking and replacing by in Lemma 1, we obtain where By Lemmas 5 and 6, we know that and are the -eigenvalues of the strongly quotient graph with multiplicity greater than or equal to and , respectively. Therefore, considering (8) we obtain Observe that Hence we get the result.

Remark 8. In [19] Zaferani obtained the following upper bound for the distance energy of : The upper bound (14) is better than the upper bound (20). Using the Arithmetic-Geometric Mean Inequality, we can easily see that Considering this and the upper bound (14), we arrive at which is the upper bound (20).

Using Theorem 7 and Lemma 3, we can give the following result.

Corollary 9. Let be a connected strongly quotient graph () with vertices and maximum edges and let , where denotes the diameter of . Then where .

4. Bounds on Distance Estrada Index of CSQG

In this section, we will use similar ideas as in [22, 2427] to obtain some bounds for , where is . These bounds are based on the distance energy and several other graph invariants.

Theorem 10. Let be a connected strongly quotient graph () with vertices and maximum edges . Then

Proof. Using the Arithmetic-Geometric Mean Inequality, we get From (7) and Lemmas 5 and 6, we have Employing (25) and (26), we conclude that This completes the proof.

Theorem 11. The distance Estrada index and the distance energy of with vertices and maximum edges satisfy the following inequalities:

Proof. Lower bound: Using Lemmas 5 and 6 and the inequalities and , we obtain From (26), we get Hence the lower bound (28).
Upper bound: Considering which monotonically increases in the interval , we obtain This completes the proof.

Remark 12. In [28] the following result was obtained for connected -graphs Since the function monotonically increases in the interval , we conclude that the upper bound (29) is better than the upper bound (33) for of with vertices and maximum edges .

Theorem 13. The distance Estrada index and the distance energy of with vertices and maximum edges satisfy the following inequality: where and is the diameter of .

Proof. From (32) and Lemma 2, we get By Lemmas 5 and 6, we know that and are the -eigenvalues of the strongly quotient graph with multiplicity greater than or equal to and , respectively. These imply that Therefore, It is easy to see that the function monotonically increases in the interval . Then by Lemma 4, we obtain From (36) and Lemma 4, we also have that is better than the upper bound (34).

From Theorem 13 and Lemma 3, we can give the following result.

Corollary 14. Let be a connected strongly quotient graph with vertices and maximum edges and let , where denotes the diameter of . Then where .

Remark 15. In [28] the following result was obtained for a connected -graph Since the functions and monotonically increase in the intervals and , respectively, we conclude that the upper bound (34) is better than the upper bound (42) for with vertices and maximum edges .

Acknowledgment

The authors thank the referees for their helpful comments and suggestions concerning the presentation of this paper.

References

  1. I. Gutman and B. Furtula, Distance in Molecular Graphs-Theory and Applications, vol. 12-13, University of Kragujevac, Kragujevac, Serbia, 2012.
  2. Z. Mihalić, D. Veljan, D. Amić, S. Nikolić, D. Plavšić, and N. Trinajstić, “The distance matrix in chemistry,” Journal of Mathematical Chemistry, vol. 11, no. 1, pp. 223–258, 1992. View at Publisher · View at Google Scholar
  3. F. Buckley and F. Harary, Distance in Graphs, Addison-Wesley, Redwood City, CA, USA, 1990. View at MathSciNet
  4. D. M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs, Johann Ambrosius Bart Verlag, Heidelberg, Germany, 3rd edition, 1995. View at MathSciNet
  5. I. Gutman and M. Medeleanu, “On the structure-dependence of the largest eigenvalue of the distance matrix of an alkane,” Indian Journal of Chemistry, vol. 37, pp. 569–573, 1998.
  6. G. Indulal, “Sharp bounds on the distance spectral radius and the distance energy of graphs,” Linear Algebra and its Applications, vol. 430, no. 1, pp. 106–113, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. G. Indulal and I. Gutman, “On the distance spectra of some graphs,” Mathematical Communications, vol. 13, no. 1, pp. 123–131, 2008. View at Zentralblatt MATH · View at MathSciNet
  8. P. Křivka and N. Trinajstić, “On the distance polynomial of a graph,” Applied Mathematics, vol. 28, no. 5, pp. 357–363, 1983. View at Zentralblatt MATH · View at MathSciNet
  9. B. Zhou, “On the largest eigenvalue of the distance matrix of a tree,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 58, no. 3, pp. 657–662, 2007. View at Zentralblatt MATH · View at MathSciNet
  10. B. Zhou and A. Ilić, “On distance spectral radius and distance energy of graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 64, no. 1, pp. 261–280, 2010. View at MathSciNet
  11. B. Zhou and N. Trinajstić, “On the largest eigenvalue of the distance matrix of a connected graph,” Chemical Physics Letters, vol. 447, no. 4–6, pp. 384–387, 2007.
  12. B. Zhou and N. Trinajstić, “Further results on the largest eigenvalues of the distance matrix and some distance based matrices of connected (molecular) graphs,” Internet Electronic Journal of Molecular Design, vol. 6, pp. 375–384, 2007.
  13. G. Indulal, I. Gutman, and A. Vijayakumar, “On distance energy of graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 60, no. 2, pp. 461–472, 2008. View at Zentralblatt MATH · View at MathSciNet
  14. I. Gutman, “The energy of a graph,” Berichte der Mathematisch-Statistischen Sektion im Forschungszentrum Graz, vol. 103, pp. 1–22, 1978. View at Zentralblatt MATH · View at MathSciNet
  15. I. Gutman, “The energy of a graph: old and new results,” in Algebraic Combinatorics and Applications, pp. 196–211, Springer, Berlin, Germany, 2001. View at Zentralblatt MATH · View at MathSciNet
  16. X. Li, Y. Shi, and I. Gutman, Graph Energy, Springer, New York, NY, USA, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  17. H. S. Ramane, D. S. Revankar, I. Gutman, S. B. Rao, B. D. Acharya, and H. B. Walikar, “Bounds for the distance energy of a graph,” Kragujevac Journal of Mathematics, vol. 31, pp. 59–68, 2008. View at Zentralblatt MATH · View at MathSciNet
  18. A. Ilić, “Distance spectra and distance energy of integral circulant graphs,” Linear Algebra and its Applications, vol. 433, no. 5, pp. 1005–1014, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. R. K. Zaferani, “A note on strongly quotient graphs,” Bulletin of the Iranian Mathematical Society, vol. 34, no. 2, pp. 97–144, 2008. View at Zentralblatt MATH · View at MathSciNet
  20. Ş. B. Bozkurt, A. D. Güngör, and B. Zhou, “Note on the distance energy of graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 64, no. 1, pp. 129–134, 2010. View at MathSciNet
  21. E. Estrada, “Characterization of 3D molecular structure,” Chemical Physics Letters, vol. 319, no. 5-6, pp. 713–718, 2000. View at Publisher · View at Google Scholar
  22. J. A. de la Peña, I. Gutman, and J. Rada, “Estimating the Estrada index,” Linear Algebra and its Applications, vol. 427, no. 1, pp. 70–76, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. Ş. B. Bozkurt, C. Adiga, and D. Bozkurt, “On the energy and Estrada index of strongly quotient graphs,” Indian Journal of Pure and Applied Mathematics, vol. 43, no. 1, pp. 25–36, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  24. G. H. Fath-Tabar, A. R. Ashrafi, and I. Gutman, “Note on Estrada and L-Estrada indices of graphs,” Bulletin Classe des Sciences Mathématiques et Naturelles Sciences Mathématiques, no. 34, pp. 1–16, 2009. View at MathSciNet
  25. I. Gutman, H. Deng, and S. Radenković, “The Estrada index : an updated survey,” D. Cvetković and I. Gutman, Eds., pp. 155–174, Selected Topics on Applications of Graph Spectra, Belgrade, Serbia, 2011.
  26. J. P. Liu and B. L. Liu, “Bounds of the Estrada index of graphs,” Applied Mathematics Journal of Chinese Universities, vol. 25, no. 3, pp. 325–330, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. B. Zhou, “On Estrada index,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 60, no. 2, pp. 485–492, 2008. View at Zentralblatt MATH · View at MathSciNet
  28. A. D. Güngör and Ş. B. Bozkurt, “On the distance Estrada index of graphs,” Hacettepe Journal of Mathematics and Statistics, vol. 38, no. 3, pp. 277–283, 2009. View at Zentralblatt MATH · View at MathSciNet
  29. C. Adiga, R. K. Zaferani, and M. Smitha, “Strongly quotient graphs,” South East Asian Journal of Mathematics and Mathematical Sciences, vol. 5, no. 2, pp. 119–127, 2007. View at MathSciNet
  30. C. Adiga and R. K. Zaferani, “On colorings of strongly multiplicative and strongly quotient graphs,” Advanced Studies in Contemporary Mathematics, vol. 13, no. 2, pp. 171–182, 2006. View at Zentralblatt MATH · View at MathSciNet
  31. B. Zhou, I. Gutman, and T. Aleksić, “A note on Laplacian energy of graphs,” MATCH Communications in Mathematical and in Computer Chemistry, vol. 60, no. 2, pp. 441–446, 2008. View at Zentralblatt MATH · View at MathSciNet