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

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).


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 spectrumbased invariants of these matrices.
Let  be a connected graph with  vertices and  edges and let the vertices of  be labeled as V 1 , V 2 , . . ., V  .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 V  and V  of .The diameter of the graph , denoted by diam(), 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  1 ≥  2 ≥ ⋅ ⋅ ⋅ ≥   .For more details on -eigenvalues, especially on  1 , see [3][4][5][6][7][8][9][10][11][12].
This concept was motivated by the ordinary graph energy which is defined as the sum of absolute values of ordinary graph eigenvalues [14][15][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][22][23][24][25][26][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  :  () → {1, 2, . . ., } . ( We define the quotient function by if  joins V and .Note that for any  ∈ (), 0 <   () < 1.
A graph with  vertices is called a strongly quotient graph if its vertices can be labeled 1, 2, . . .,  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 SQG and CSQG 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 DEE() as well as some relations between DEE() and   () where  is CSQG.We present some bounds for   () and DEE() of CSQG whose diameter does not exceed two.We also show that our results improve most of the results obtained in [19,28] for CSQG.

Preliminaries
In this section, we give some lemmas which will be used in our main results.
Lemma 4 (see [28]).Let  be a connected (, )-graph and diam() be the diameter of .Then The equality holds in (10) if and only if  ≅   .
Lemma 5 (see [19]).If  is a SQG, then −1 is a -eigenvalue of  with multiplicity greater than or equal to || = , where Lemma 6 (see [19]).If  is a SQG, then −2 is a -eigenvalue of  with multiplicity greater than or equal to  where

Bounds on Distance Energy of CSQG
In this section, we will present a better upper bound and a new lower bound for   () where  is CSQG with eigenvalues  1 ,  2 , . . .,   .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 where By Lemmas 5 and 6, we know that −1 and −2 are the eigenvalues of the strongly quotient graph  with multiplicity greater than or equal to  and , respectively.Therefore, considering Observe that Hence we get the result.
Remark 8.In [19] Zaferani obtained the following upper bound for the distance energy of CSQG: 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.

Bounds on Distance Estrada Index of CSQG
In this section, we will use similar ideas as in [22,[24][25][26][27] to obtain some bounds for DEE(), where  is CSQG.These bounds are based on the distance energy   () and several other graph invariants.
Theorem 10.Let  be a connected strongly quotient graph (CSQG) with  > 3 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 DEE() and the distance energy   () of CSQG with  > 3 vertices and maximum edges  satisfy the following inequalities: Proof.Lower bound: Using Lemmas 5 and 6 and the inequalities   ≥  and   ≥ 1 + , we obtain From ( 26), we get Hence the lower bound (28).
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 DEE() of CSQG with  > 3 vertices and maximum edges .
Theorem 13.The distance Estrada index DEE() and the distance energy   () of CSQG with  > 3 vertices and maximum edges  satisfy the following inequality: where  = ( − 1)diam 2 () −  − 4 and diam() is the diameter of .
Proof.From (32) and Lemma 2, we get By Lemmas 5 and 6, we know that −1 and −2 are the eigenvalues of the strongly quotient graph  with multiplicity greater than or equal to  and , respectively.These imply that Therefore, that is better than the upper bound (34).