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

Ş. Burcu Bozkurt,¹ Chandrashekara Adiga,² and Durmuş Bozkurt¹

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 [3–12].

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 [14–16]. 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 [21–27]. 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, 24–27] 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.

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