THE NORMALIZED LAPLACIAN ESTRADA INDEX OF GRAPHS

Abstract

Suppose G is a simple graph. The ${\ell}$-eigenvalues ${\delta}_1$, ${\delta}_2$,..., ${\delta}_n$ of G are the eigenvalues of its normalized Laplacian ${\ell}$. The normalized Laplacian Estrada index of the graph G is dened as ${\ell}EE$ = ${\ell}EE$(G) = ${\sum}^n_{i=1}e^{{\delta}_i}$. In this paper the basic properties of ${\ell}EE$ are investigated. Moreover, some lower and upper bounds for the normalized Laplacian Estrada index in terms of the number of vertices, edges and the Randic index are obtained. In addition, some relations between ${\ell}EE$ and graph energy $E_{\ell}$(G) are presented.

1. Introduction

Let G = (V,E) be a simple graph with n vertices and m edges. The eigenvalues of the adjacency matrix A(G) are called the eigenvalues of G and form the spectrum of G. Suppose {λ1, λ2, . . . , λn} is the spectrum of G such that λ1 ≤ λ2 ≤ · · · ≤ λn. If G has exactly s distinct eigenvalues δ1, . . . , δs and the multiplicity of δ1 is t1, 1 ≤ i ≤ s, then we use the following compact form

for the spectrum of G.

The Estrada index of the graph G is defined as This graph invariant was introduced by Ernesto Estrada, which has noteworthy chemical applications, see [12,13,14] for details. We encourage the interested readers to consult [2,11,16] for the mathematical properties of Estrada index.

The Laplacian matrix of G is defined as L(G) = D(G) − A(G), where A(G) and D(G) are the adjacency and diagonal matrices of G, respectively. If 0 = μ1 ≤ μ2 ≤ · · · ≤ μn are the Laplacian eigenvalues of G, then the Laplacian Estrada index, L-Estrada index for short, of G is defined as the sum of the terms eμi, 1 ≤ i ≤ n. This quantity is denoted by LEE (G). There exists a vast literature that studies the L-Estrada index of graphs. We refer the readers to [15,19,24] for more information.

The normalized Laplacian matrix ℓ(G) = [ℓi,j]n×n is defined as:

The normalized Laplacian eigenvalues or ℓ−spectrum of G are denoted by 0 = δ1 ≤ δ2 ≤ · · · ≤ δn. The multiplicity of δ1 = 0 is equal to the number of connected components of G. Define φ(G, δ) = det (δIn − ℓ(G)), where In is the unit matrix of order n. This polynomial is called the normalized Laplacian characteristic polynomial. The basic properties of the normalized Laplacian characteristic polynomial. The basic properties of the normalized Laplacian eigenvalues can be found in [8,9]. The normalized Laplacian eigenvalues of an n−vertex connected graph G satisfying the following elementary conditions: where R−1(G) is Randic index of G, see [6,8,9] for details.

We now define the normalized Laplacian Estrada index, simply called ℓ−Estrada index, of G by the following equation:

From the power-series expansion of ex, we have:

where we assumed that 00 = 1.

We now introduce some notation that will be used throughout this paper. The complete graph on n vertices is denoted by Kn. The line graph l(G) of a graph G is another graph l(G) that represents the adjacencies between edges of G. In a graph theoretical language V (l(G)) = E(G) and two edges of G are adjacent in l(G) if they have a common vertex. Suppose denotes the complement of G. For two graphs G and H, G ∪ H is the disjoint union of G and H. The join G + H is the graph obtained from G ∪ H by connecting all vertices from V (G) with all vertices from V (H). If G1,G2, . . . ,Gk are graphs with mutually disjoint vertex sets, then we denote G1 + G2 + · · · + Gk by In the case that G1 = G2 = . . . = Gk = G, we denote

The following results are crucial throughout this paper.

Lemma 1.1 (See [9] for details). Let G be a graph of order n ≥ 2 that contains no isolated vertices. We have

i) If G is connected with m edges and diameter D, then

ii) with equality if and only if G is the complete graph on n vertices.

Lemma 1.2 ([21] Theorems 2.2 and 2.3). Let G be a graph of order n with no isolated vertices. Suppose G has minimum vertex degree equal to dmin and maximum vertex degree equal to dmax. Then Equality occurs in both bounds if and only if G is a regular graph.

Lemma 1.3. Let G be an n−vertex graph. Then δ2 = · · · = δn if and only if

Proof. We know that δ1 = 0. Suppose that δ2 = · · · = δn. If G is connected on n ≥ 3 vertices, then by [7, Corollary 2.6.4] G has exactly two distinct ℓ−eigenvalues if and only if G is the complete graph. If G is not connected, then δ2 = 0 and if δi = 0 and δi+1 ≠ 0 then by [9, Lemma 1.7 (iv)], G has exactly i connected components. So, all Laplacian eigenvalues are equal to zero, which obviously implies that

 

2. Examples

In this section, the normalized Laplacian Estrada index of some well-known graphs are computed.

Example 2.1. In this example the normalized Laplacian Estrada index of complete and cocktail-party graphs are computed. We begin with the complete graph. The normalized Laplacian spectrum of Kn and cocktail-party graph are computed as follows:

Example 2.2. The normalized Laplacian spectrum of the cycle Cn consists of

Suppose

Example 2.3. The normalized Laplacian spectrum of n−vertex path Pn consists of

Therefore, ℓEE(Pn) ≈ 0.753004179 + 3.441523869n, for large n.

Consider the Petersen graph P on 10 vertices. Then the normalized Laplacian spectrum of P is

Example 2.4. Take the star graph and add a new edge to each of its n vertices to get a star-like graph T2t+1 with n = 2t+1 vertices. By [8], the ℓ−eigenvalues of a star-like graph are as follows:

FIGURE 1.The Star-Like Graph T2t+1

Therefore, ℓEE(G) = 1 + e + e2 + e(n − 3) cosh

Example 2.5. Suppose G is a m−petal graph on n = 2m+1 vertices, V (G) = {v0, v1, . . . , v2m} and E(G) = {v0vi, v2i-1v2i}, for i > 1.

By [9], G has ℓ−eigenvalues 0, with multiplicity m−1, and with multiplicity We now generalize this graph as follows: Fix s,m ≥ 2 and let H = {u}+(sKm), see Figure 2 for an illustration.

By [8], The ℓ−eigenvalues of H are 0, with multiplicity s − 1 and with multiplicity s(m−1)+1. Then,

FIGURE 2.The Generalized Petal Graph.

Example 2.6. Let G be the graph constructed as follows. Fix m ≥ 1. Take the vertex set to be {u1, u2, u3} ∪ V1 ∪ V2 ∪ V3, where each Vi is a set of m vertices. Then G has exactly 3(m + 1) vertices. Define the edge set of G by

see Figure 3. By [7], the ℓ−eigenvalues of G are 0, with multiplicity 2 and with multiplicity 3m. Hence,

FIGURE 3.The Generalized Triangle-Petal Graph.

Example 2.7. The hypercube graph Qn is a regular graph with 2n vertices, which correspond to the subsets of an n−element set. Two vertices A and B are joined by an edge if and only if A can be obtained from B by adding or removing a single element. The ℓ−eigenvalues of the hypercube Qn are with multiplicity for 0 ≤ i ≤ n. So,

Example 2.8. The wheel graph on n+1 vertices is defined by Wn = Cn +K1. Thus, the normalized Laplacian spectrum is

Thus,

Define

Example 2.9. A Möbius ladder Ln of order 2n is a graph obtained by introducing a twist in a 3−regular prism graph of order n that is isomorphic to the circulant graph, see Figure 4.

FIGURE 4.The Möbuis Ladder Graph.

In this example the normalized Laplacian Estrada index of a Möbius graph is computed. By [10], the normalized Laplacian eigenvalues of Ln are where 0 ≤ i ≤ 2n − 1. So,

Note that,

Since, for large n.

 

3. The ℓ−Estrada Index of Graphs

This section is concerned with the use of algebraic techniques in the study of the normalized Estrada index of graphs. We begin with the following simple theorem:

Theorem 3.1. Let G be a connected graph with n vertices. Then ℓEE(G) > ne.

Proof. By Arithmetic-Geometric mean inequality [18], we have:

with equality if and only if for all 1 ≤ i, j ≤ n, eδi = eδi if and only if δi = δj. This implies that all δi, s are zero. This contradicts the fact that G is connected.

Theorem 3.2. Let G be a graph with n vertices and c connected components. Then, Equality holds if and only if G is a union of copies of cKs, for some fixed integers s.

Proof. Using a similar method as in [3, Theorem 3], we obtain δ1 = · · · = δc = 0 and δc+1 + · · · + δn = n. Therefore,

where the last inequality is obtained by applying the Arithmetic-Geometric mean inequality. Suppose G = cKs, s ≥ 2. Then, n = cs, and the normalized Laplacian spectrum of G is as follows: Further,

This shows that the equality holds for G. Conversely, let equality hold for G. Then all of non-zero normalized Laplacian eigenvalues of G must be mutually equal. Then, the normalized Laplacian spectrum of the graph H is 0, δ with multiplicity s − 1, where δ > 0 and s is a positive integer. Therefore, H = Ks, and then G = cKs, as desired.

Theorem 3.3. If G is a connected r−regular graph with n vertices, then with equality if and only if

Proof. The ℓ−spectrum of G is 0, for 2 ≤ i ≤ n. Then By arithmetic-geometric mean inequality, we get

where the last equality follows from Therefore, with equality if and only if λ2 = · · · = λn. By assumption, this is equivalent to

Theorem 3.4. If G is an r−regular bipartite graph, then

Proof. The ℓ−eigenvalues of G are 0, Thus,

where the last equality follows from Since G is bipartite, the eigenvalues of G are symmetric around zero. The equality is attained if and only if λ1 = · · · = λn and this is equivalent to which is impossible.

Theorem 3.5. Let G be a connected with n ≥ 2 vertices, m edges and diameter D. Then

Proof. Since G is connected, δ1 = 0 and δ2, δn > 0. Then,

Define x, y > 0. Then we have:

Moreover, if fx = fy = 0 then (n − 2)x + y = n and so If then fxx > 0 and

From the above, we conclude that f(x, y) has a minimum at and that the minimum value is Hence f is an increasing function for x > 0. By Lemma 1.1(i), Thus,

proving the result.

Theorem 3.6. If G is an r−regular graph with n vertices, then

with equality if and only if In particular, for r−regular graphs, if and only if r = 2.

Proof. By [4, Theorem 3.8], the eigenvalues of l(G) are −2 with multiplicity and λi(G) + r − 2 for 1 ≤ i ≤ n. Since the line graph of G is (2r − 2)−regular, and μi(l(G)) = 2r − 2 − λi(l(G)) for 1 ≤ i ≤ n, the normalized Laplacian eigenvalues of l(G) are Thus, we have:

From [24, Lemma 1.2] it follows that the above equality holds if and only if G is an empty graph.

Corollary 3.7. Let l(G) = l1(G) and lk+1(G) = l(lk(G)). If G is r−regular then where lk(G) is rk−regular with nk vertices,

Corollary 3.8. If G is 2−regular and bipartite, then

A fullerene graph of order n is a cubic 3−connected planar graph with exactly 12 pentagonal faces and -10 hexagonal faces.

Corollary 3.9. If Fn is an n−vertex fullerene graph, then

Consider G is r−regular graph with n−vertex and m-edges, and the eigenvalues of G are r = λ1(G), λ2(G), . . . , λn(G ). A para-line graph of G, denoted by C(G), is defined as a line graph of the subdivision graph S(G) (i.e., S(G) is the graph obtained from G by inserting a vertex to every edge of G.) of G. The para-line graph has also been called the clique-inserted graph. Note that para-line graph is r−regular and the number of vertices of C(G) equals nr. The eigenvalues of the para-line graph C(G) of G are for 1 ≤ i ≤ n, −2, with multiplicity m − n, and 0, with multiplicity m − n, see [22,23] for details.

Theorem 3.10. Let G be a r−regular graph with n vertices and m edges. Then

Proof. By above discussion, the normalized Laplacian eigenvalues of the paraline graph C(G) of G are

By definition,

In the other hand,

where the last equality follows from Therefore,

Corollary 3.11. Let C0(G) = G, Ck(G) = C(Ck−1(G)), k ≥ 1. Then where Ck(G) is r−regular with vertices for k ≥ 0.

Theorem 3.12. Let G be an r−regular graph. Then

Equality holds if and only if G is an empty graph.

Proof. By [10, Theorem 2.6], if the spectrum of G contains r = λ1, λ2, . . . , λn, then the spectrum of is n−r−1 and −1−λi, where 2 ≤ i ≤ n. Since μi = r−λi and complement of G is (n−r−1)−regular, the normalized Laplacian eigenvalues of where 2 ≤ i ≤ n. Thus,

Clearly, equality holds if and only if G is an empty graph.

Theorem 3.13. Let G1 and G2 be r− and s−regular graphs on n and m vertices, respectively. Suppose 0 = δ1(G1) ≤ δ2(G1) ≤ · · · ≤ δn(G1) ≤ 2 are the ℓ−eigenvalues of G1 and 0 = δ1(G2) ≤ δ2(G2) ≤ · · · ≤ δn(G2) ≤ 2 are the ℓ−eigenvalues of G2. Then

with equality if and only if

Proof. From [6, Theorem 12], the normalized Laplacian eigenvalues of G1 + G2 are as follows:

Hence,

where the last equality follows from δ1(G1) = 0 and δ1(G2) = 0. The equality is attained if and only if δj(G1) = 0, 2 ≤ i ≤ n, and δj (δ2) = 0, 2 ≤ j ≤ m. So, This completes the proof.

Apply Theorems 3.12 and 3.13 to evaluate the ℓ−Estrada indices of the complete bipartite graphs, star graphs, CPn +2K1 and Kn−2 +2K1. Start with the complete bipartite graph Kn,m. We have:

Corollary 3.14. If Gj, 1 ≤ j ≤ k, is an r−regular n−vertex graph, then

Corollary 3.15. If G is an r−regular graph n−vertex graph, then

Theorem 3.16. If G is connected graph with n vertices, then

Proof. Using a similar method as [15, Proposition 7], we have

resulting in the upper bound. If then δi = 0, where 2 ≤ i ≤ n. Thus, Obviously, the right equality is impossible. On the other hand, and so, by the arithmetic-geometric inequality

By means of a power-series expansion, we get

Therefore, ℓEE(G)2 = n(n−1)e2+4R−1(G) +5n. This implies the lower bound. If then δi = 0 for 2 ≤ i ≤ n. Thus, The left equality is clearly impossible, proving the result.

Theorem 3.17. If G is a connected graph with n > 2 vertices, then

Proof. Using a similar method as in [19, Proposition 3.3], one can observe that for with equality for all k ≥ 2 if and only if Then

In Theorem 3.16, it was shown that 2 Thus,

Note that ex ≥ (1+x), so if n > 2 then n(n−1)e2−6n+4 ≥ 3n(n−1)−6n+4 ≥ 0. Therefore,

Since the graph is connected, the equality can not be attained.

Theorem 3.18. If G is a connected graph with n vertices, then

Proof. By definition,

and by Lemma 1.2, Also, the equality occurs if only if which is impossible.

Corollary 3.19. If G is an r−regular connected graph with n vertices, then

Theorem 3.20. If G is connected graph with n vertices, then

Proof. Recall that Using a similar method as [24, Proposition 3.1], for an integer It is easily seen that

with equality if and only if at most one of δ1, δ2, . . . , δn is non-zero, or equivalently which is impossible.

Theorem 3.21. If G is connected graph with n vertices, then

with equality if and only if

Proof. Using a similar method as given in [19, Proposition 3.4] and by an inequality from [16, p. 26], where a1, a2, . . . , ap are non-negative numbers and m ≤ k with m, k ≠ 0, we have:

Equality is attained if and only if a1 = a2 = · · · = ap. In above inequality, we substitute m = 2, p = n − 1, ai = δi, 2 ≤ i ≤ n and k ≥ 2. Then we have:

which is an equality for k = 2 whereas equality holds for k ≥ 3 if and only if δ2 = · · · = δn. By Lemma 1.3, this is equivalent to Since G is a connected graph, Clearly,

with equality if and only if the lower bound for above is attained for k ≥ 3, if and only if

 

4. Bounds for the ℓ−Estrada Index

We recall that the normalized Laplacian energy of the graph G is defined as [8]. In this section, the relationship between the ℓ−Estrada index and the normalized Laplacian energy of graphs are investigated.

Theorem 4.1. If G is connected, then ℓEE(G) < e(n − 1 + eEℓ(G)).

Proof. By definition, we have

with equality if and only if if and only if δi = 0, 1 ≤ i ≤ n, if and only if G is an empty graph with n vertices, which is impossible.

In [5], the authors introduced the notion of the Randić matrix of a graph G as R(G) = [Ri,j]n×n, where

The Randić energy of G is defined by where τi's are the eigenvalues of Randić matrix R(G).

Corollary 4.2. If G is connected then ℓEE(G) < e(n − 1 + eER(G)).

Proof. The proof is follows from [5, Theorem 2] and Theorem 4.1.

Theorem 4.3. If G is a connected graph with n vertices, then

Proof. In the proof of Theorem 3.18, the following inequality is proved:

On the other hand, by definition of the normalized Laplacian energy,

Thus,

We now apply Lemma 1.2, to get The equality is attained if and only if which is impossible.

Corollary 4.4. If G is an r−regular n−vertex graph, then

Theorem 4.5. Let p, q and s be, respectively, the numbers of normalized Lapla-cian eigenvalues which are greater than, equal to, and less than 1. Then

Proof. Let δ1, . . . , δp be the normalized Laplacian eigenvalues of G greater than 1, and δn-s+1, . . . , δn be the normalized Laplacian eigenvalues less than 1. Since the sum of normalized Laplacian eigenvalues of a connected graph G is n and

by the arithmetic-geometric mean inequality, we have:

and for eigenvalues equal to 1, Now, the result is obtained by combining these inequalities.