Generalization for Estrada Index A. Dilek G¨ ung¨ or
∗ (1),
, A. Sinan ¸ Cevik
Firat Ate¸ s (1)
(2)
(1)
, Eylem G. Karpuz
(2)
,
and I. Naci Cang¨ ul†,(3)
Sel¸cuk University, Department of Mathematics,
Faculty of Science, Campus, 42075, Konya - Turkey
[email protected] and
[email protected] (2)
Balikesir University, Department of Mathematics,
Faculty of Art and Science, Cagis Campus, 10145, Balikesir - Turkey
[email protected] and
[email protected] (3)
Uludag University, Department of Mathematics,
Faculty of Science and Art, G¨or¨ ukle Campus, 16059, Bursa - Turkey
[email protected]
Abstract In this paper the Estrada index of Hermite matrix is firstly defined and investigated. In fact this is a natural generalization of Estrada, distance Estrada and Laplacian Estrada indices. Thus all properties about them can be handled by this new index. AMS Subject Classification: 15C12; 05C50; 15A36; 15A42. Keywords and Phrases: Estrada index, Laplacian Estrada index, Bounds.
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Introduction and Preliminaries A graph G with n vertices and m edges is called (n, m)-graph if it is simple (i.e., if it
is an undirected graph with no loops and no multiple edges). Throughout this paper, all ∗ †
Corresponding author This author is supported by The Research Fund of Uludag University, Project Nos: 2006-40, 2008-31
and 2008-54
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graphs will be concerned as a (n, m)-graph. Let A = A(G) be the adjacency matrix of G, and let λ1 , λ2 , · · · , λn be its eigenvalues. By [1], it is known that these eigenvalues form the spectrum of the graph G. Let G be connected graph on the vertex set V = {v1 , v2 , · · · , vn }. Then the distance matrix D = D(G) of G is defined as its (i, j)-entry is equal to dG (vi , vj ), denoted by dij , the distance (in other words, the lenght of the shortest path) between the vertices vi and vj of G. Let ρ1 , ρ2 , · · · , ρn be the eigenvalues of D(G). Moreover let L = L(G) be the Laplacian matrix of G (formally it is denoted by L(G) = D(G) − A(G)), and let µ1 , µ2 , · · · , µn be its eigenvalues. These eigenvalues form the Laplacian spectrum of the graph G (see [12, 13]). Since A(G), L(G) and D(G) are real symmetric matrices, their eigenvalues are real numbers and so we can order them as λ1 ≥ λ2 ≥ · · · ≥ λn , µ1 ≥ µ2 ≥ · · · ≥ µn and ρ1 ≥ ρ2 ≥ · · · ≥ ρn . These eigenvalues are shortly called A-eigenvalues, L-eigenvalues and D-eigenvalues, respectively. The fundamental properties of graph eigenvalues can be found in the book [1]. In fact, by considering adjacency and Laplacian spectrums n X of G, the energy of G ([7]) is defined by E(G) = |λi | while the Laplacian energy of n X
i=1
2m |. Moreover the Estrada index of a simple n i=1 n X connected graph G is defined by EE(G) = eλi , where λ1 ≥ λ2 ≥ · · · ≥ λn are the G ([14]) is defined by LE(G) =
|µi −
i=1
A-eigenvalues as depicted above (see, for instance, [4]). Denoting by Mk = Mk (G) to the n X k-th moment of the graph G, we get Mk = Mk (G) = (λi )k , and recalling the powerseries expansion of ex , we have EE(G) =
∞ X Mk
i=1
. We may refer [6] to the reader for a list k! k=0 of references that study on the useage of Estrada index in chemistry and physics (see also [2, 3, 8, 11]). Further, we should recall another graph-spectrum-based invariant, namely n X 2m the Laplacian Estrada index, which is defined by LEE(G) = eµi − n . It is known that i=1
([9]) LEE(G) and EE(G) have a number of common properties. Finally we recall that n X the distance Estrada index ([6]) for a (n, m)-graph G is defined by DEE(G) = eρi , where ρ1 ≥ ρ2 ≥ · · · ≥ ρn are the D-eigenvalues.
i=1
Now we will focus on to generalize the concept “Estrada index” to the Hermite matrix A. Thus one can deal with all results about Estrada, distance Estrada and Laplacian Estrada indices for a graph G. Therefore let us consider a Hermite matrix, i.e., a square matrix A = (aij ) such that A = A∗ where A∗ is the conjugate transpose (aji ) of A. It is
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well known that the eigenvalues of Hermite matrix A are real, and so they can be labelled according to decreasing size. In [10], Liu et.al. extended the concepts of the energy and the Laplacian energy of the graph G to Hermite matrix as in the following: Let A be a Hermite matrix of order n and let γ1 ≥ γ2 ≥ · · · ≥ γn be the eigenvalues of A. Then the energy of A is defined by HE(A) =
n X
|γi −
i=1
tr(A) |, n
(1)
where tr(A) denotes the trace of A. Considering this and EE(G), we can extend the concept Estrada index to the Hermite matrix as follows: Definition 1.1 For a Hermite matrix A of order n, let γ1 ≥ γ2 ≥ · · · ≥ γn be the eigenvalues of A. Then the Estrada index of Hermite matrix is defined by HEE(A) =
n X
eγi −
tr(A) n
.
(2)
i=1
It is clear that if A is Hermitian, then A −
tr(A) I n
is also Hermitian, where I denotes
the identity matrix of order n. So the eigenvalues ζ1 , ζ2 , · · · , ζn of A −
tr(A) I n
can also be
labelled as decreasing order. We clearly have ζi + tr(A) = γi or, equaivalently, ζi = γi − tr(A) . n n In addition, again, denoting by Mk = Mk (A) to the k-th moment of the graph G, we get n X Mk = Mk (A) = (ζi )k . Therefore Eq. (2) can be rewritten as i=1
HEE(A) =
n X
ζi
e
or
HEE(A) =
i=1
∞ X Mk k=0
k!
.
(3)
In this paper we will investigate the general properties for the Estrada index of Hermite matrix HEE(A) in order to a well understanding of the definition and properties of it.
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Bounds for HEE(A) By considering the definition of HEE(A) defined in (2) and (3), we will present some
bounds on it. Theorem 2.1 Let A = (aij ) be a Hermite matrix. Then HEE(A) ≥ n, and equality holds if and only if A = kI, for any real constant k. Proof. From the inequality between the arithmetic and geometric means, we have v u n p √ uY Pn HEE(A) n n n ζ t i ≥ e = e i=1 ζi = e0 = 1. n i=1 3
Actually equality holds if and only if eζi = eζj , for all 1 ≤ i, j ≤ n, that is if and only if ζi = ζj . This implies that all ζi ’s are zero. This means that, for any real constant k, A = kI, as required. We then have the following lower bound on Estrada index of graphs. Corollary 2.2 [5] If A is the adjacency matrix of the simple graph G, then tr(A) = 0. Thus EE(G) ≥ n, and equality holds if and only if G is the empty graph. The proof of the following lemma can be found in [10]. n X
Lemma 2.3 Let A = (aij ) be a Hermite matrix, and let S =
|aij |2 where |aij | is the
i,j=1
modulo of the complex entry aij . Then we have n X
ζi = 0,
i=1
n X
ζi2 = S −
i=1
n X
tr(A)2 , n
γi = tr(A) and
i=1
n X
γi2 = S.
i=1
Theorem 2.4 Let A = (aij ) be a Hermite matrix. Then r q tr(A)2 tr(A)2 2 n + 2(S − ) ≤ HEE(A) ≤ n − 1 + e S− n . n
(4)
Equality is attained on both sides of (4) if and only if A = kI, for any real constant k. Proof. The Lower Bound : By a direct calculation from the first equality in Eq. (3), HEE 2 (A) =
n X
e2ζi + 2
i=1
X
eζi eζj .
i
