2.TheKnownUpperBoundsfortheLargestLaplacianEigenvalueofWeightedGraphs 1.Introduction SezerSorgun BoundsfortheLargestLaplacianEigenvalueofWeightedGraphs ResearchArticle

Volume 2013, Article ID 520610,8pages http://dx.doi.org/10.1155/2013/520610

Research Article

Bounds for the Largest Laplacian Eigenvalue of Weighted Graphs

Sezer Sorgun

Department of Mathematics, Faculty of Sciences and Arts, Nevs¸ehir University, 50300 Nevs¸ehir, Turkey

Correspondence should be addressed to Sezer Sorgun; [email protected] Received 29 November 2012; Accepted 14 March 2013

Academic Editor: Jun-Ming Xu

Copyright © 2013 Sezer Sorgun. 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.

LetGbe weighted graphs, as the graphs where the edge weights are positive definite matrices. The Laplacian eigenvalues of a graph are the eigenvalues of Laplacian matrix of a graphG. We obtain two upper bounds for the largest Laplacian eigenvalue of weighted graphs and we compare these bounds with previously known bounds.

1. Introduction

Let𝐺 = (𝑉, 𝐸)be simple graphs, as graphs which have no loops or parallel edges such that𝑉is a finite set of vertices and𝐸is a set of edges.

A weighted graph is a graph each edge of which has been assigned to a square matrix called the weight of the edge. All the weight matrices are assumed to be of same order and to be positive matrix. In this paper, by “weighted graph” we mean

“a weighted graph with each of its edges bearing a positive definite matrix as weight,” unless otherwise stated.

The notations to be used in paper are given in the following.

Let𝐺be a weighted graph on𝑛vertices. Denote by𝑤_𝑖,𝑗 the positive definite weight matrix of order𝑝of the edge𝑖𝑗, and assume that𝑤_𝑖𝑗 = 𝑤_𝑗𝑖. We write𝑖 ∼ 𝑗if vertices𝑖and𝑗 are adjacent. Let𝑤_𝑖 = ∑_{𝑗:𝑗∼𝑖}𝑤_𝑖𝑗. be the weight matrix of the vertex𝑖.

The Laplacian matrix of a graph𝐺is defined as𝐿(𝐺) = (𝑙_𝑖𝑗), where

𝑙_𝑖,𝑗={{ {{ {

𝑤_𝑖; if 𝑖 = 𝑗,

−𝑤_𝑖𝑗; if 𝑖 ∼ 𝑗, 0; otherwise.

(1)

The zero denotes the 𝑝 × 𝑝zero matrix. Hence𝐿(𝐺)is square matrix of order𝑛𝑝. Let𝜆₁denote the largest eigenvalue

of𝐿(𝐺). In this paper we also use to avoid the confusion that 𝜌₁(𝑤_𝑖𝑗)is the spectral radius of𝑤_𝑖𝑗matrix. If𝑉is the disjoint union of two nonempty sets𝑉₁and𝑉₂such that every vertex 𝑖in𝑉₁has the same𝜌₁(𝑤_𝑖)and every vertex𝑗in𝑉₂has the same𝜌₁(𝑤_𝑗), then𝐺is called a weight-semiregular graph. If 𝜌₁(𝑤_𝑖) = 𝜌₁(𝑤_𝑗)in weight semiregular graph, then𝐺is called a weighted-regular graph.

Upper and lower bounds for the largest Laplacian eigen- value for unweighted graphs have been investigated to a great extent in the literature. Also there are some studies about the bounds for the largest Laplacian eigenvalue of weighted graphs [1–3]. The main result of this paper, contained in Section 2, gives two upper bounds on the largest Laplacian for weighted graphs, where the edge weights are positive definite matrices. These upper bounds are attained by the same methods in [1–3]. We also compare the upper bounds with the known upper bounds in [1–3]. We also characterize graphs which achieve the upper bound. The results clearly generalize some known results for weighted and unweighted graphs.

2. The Known Upper Bounds for the Largest Laplacian Eigenvalue of Weighted Graphs

In this section, we present the upper bounds for the largest Laplacian eigenvalue of weighted graphs and very useful lemmas to prove theorems.

Theorem 1 (Horn and Johnson [4]). Let𝐴 ∈ 𝑀_𝑛be Hermi- tian, and let the eigenvalues of𝐴be ordered such that 𝜆_𝑛 ≤ 𝜆_𝑛−1≤ ⋅ ⋅ ⋅ ≤ 𝜆₁. Then,

𝜆_𝑛𝑥^𝑇𝑥 ≤ 𝑥^𝑇𝐴𝑥 ≤ 𝜆₁𝑥^𝑇𝑥 (2) 𝜆_max= 𝜆₁=max

𝑥 ̸= 0

𝑥^𝑇𝐴𝑥 𝑥^𝑇𝑥 =max

𝑥^𝑇𝑥=1𝑥^𝑇𝐴𝑥 𝜆_min = 𝜆_𝑛 =min

𝑥 ̸= 0

𝑥^𝑇𝐴𝑥 𝑥^𝑇𝑥 = min

𝑥^𝑇𝑥=1𝑥^𝑇𝐴𝑥

(3)

for all𝑥 ∈C^𝑛.

Lemma 2 (Horn and Johnson [4]). Let𝐵be a Hermitian𝑛×𝑛 matrix with eigenvalues𝜆₁ ≥ 𝜆₂ ≥ ⋅ ⋅ ⋅ ≥ 𝜆_𝑛; then for any 𝑥 ∈ 𝑅^𝑛 (𝑥 ̸= 0),𝑦 ∈ 𝑅^𝑛 (𝑦 ̸= 0),

󵄨󵄨󵄨󵄨󵄨𝑥^𝑇𝐵𝑦󵄨󵄨󵄨󵄨󵄨 ≤ 𝜆¹√𝑥^𝑇𝑥√𝑦^𝑇𝑦. (4) Equality holds if and only if𝑥is an eigenvector of𝐵correspond- ing to𝜆₁and𝑦 = 𝛼𝑥for some𝛼 ∈ 𝑅.

Lemma 3 (see [1]). Let 𝐺 be a(𝜌₁(𝑤_𝑖), 𝜌₁(𝑤_𝑗))-semiregular bipartite graph of order𝑛such that the first 𝑙vertices of the same largest eigenvalue𝜌₁(𝑤_𝑖)and the remaining𝑚vertices of the same largest eigenvalue𝜌₁(𝑤_𝑗). Also let𝑥be a common eigenvector of 𝑤_𝑖𝑗 corresponding to the largest eigenvalue 𝜌₁(𝑤_𝑖𝑗)for all𝑖,𝑗, where𝑤_𝑖= ∑_{𝑘∈𝑁𝑖}𝑤_𝑖𝑘for all𝑖. Then𝜌₁(𝑤_𝑖) + 𝜌₁(𝑤_𝑗)is the largest eigenvalue of𝐿(𝐺)and the corresponding eigenvector is

(𝜌⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟₁(𝑤_𝑖)𝑥^𝑇, . . . , 𝜌₁(𝑤_𝑖)𝑥^𝑇

𝑙

, −𝜌⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟₁(𝑤_𝑗)𝑥^𝑇, . . . , −𝜌₁(𝑤_𝑗)𝑥^𝑇

𝑚

) . (5) Theorem 4 (see [1]). Let G be a simple connected weighted graph. Then

𝜆₁≤max

𝑖∼𝑗

{{ {

𝜌₁( ∑

𝑘:𝑘∼𝑖

𝑤_𝑖𝑘) + ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘)} }}

, (6) where𝑤_𝑖𝑗is the positive definite weight matrix of order p of the edge𝑖𝑗. Moreover equality holds in(6)if and only if

(i)𝐺is a weight-semiregular bipartite graph,

(ii)𝑤_𝑖𝑗 have a common eigenvector corresponding to the largest eigenvalue𝜌₁(𝑤_𝑖𝑗)for all𝑖,𝑗.

Theorem 5 (see [2]). Let G be a simple connected weighted graph. Then

𝜆₁

≤max

𝑖∼𝑗

{{ {{ {{ {{ {{ {

√√

√√

√

𝑘:𝑘∼𝑖∑ 𝜌₁(𝑤_𝑖𝑘) ( ∑

𝑟:𝑟∼𝑖𝜌₁(𝑤_𝑖𝑟) + ∑

𝑠:𝑠∼𝑘𝜌₁(𝑤_𝑘𝑠)) + ∑𝑘:𝑘∼𝑗𝜌₁(𝑤_𝑗𝑘) ( ∑

𝑟:𝑟∼𝑗𝜌₁(𝑤_𝑗𝑟) + ∑

𝑠:𝑠∼𝑘𝜌₁(𝑤_𝑘𝑠)) }} }} }} }} }} } ,

(7)

where𝑤_𝑖𝑗is the positive definite weight matrix of order p of the edge𝑖𝑗. Moreover equality holds in(7)if and only if

(i)𝐺is a bipartite semiregular graph;

(ii)𝑤_𝑖𝑗 have a common eigenvector corresponding to the largest eigenvalue𝜌₁(𝑤_𝑖𝑗)for all𝑖,𝑗.

Corollary 6 (see [2]). Let𝐺be a simple connected weighted graph where each edge weight𝑤_𝑖𝑗is a positive number. Then

𝜆₁≤max

𝑖 {√2𝑤_𝑖(𝑤_𝑖+ 𝑤_𝑖)} , (8)

where𝑤_𝑖 = (∑_{𝑘:𝑘∼𝑖}𝑤_𝑖𝑘𝑤_𝑘)/𝑤_𝑖and𝑤_𝑖is the weight of vertex𝑖.

Moreover equality holds if and only if𝐺is a bipartite regular graph.

Corollary 7 (see [2]). Let𝐺be a simple connected weighted graph where each edge weight𝑤_𝑖𝑗is a positive number. Then

𝜆₁≤max

𝑖∼𝑗 {√𝑤_𝑖(𝑤_𝑖+ 𝑤_𝑖) + 𝑤_𝑗(𝑤_𝑗+ 𝑤_𝑗)} , (9) where𝑤_𝑖 = (∑_{𝑘:𝑘∼𝑖}𝑤_𝑖𝑘𝑤_𝑘)/𝑤_𝑖and𝑤_𝑖 is the weight of vertex 𝑖. Moreover equality holds if and only if 𝐺 is a bipartite semiregular graph.

Theorem 8 (see [2]). Let G be a simple connected weighted graph. Then

𝜆₁

≤max

𝑖∼𝑗

{{ {{ {{ {

𝜌₁(𝑤_𝑖) + 𝜌₁(𝑤_𝑗) + √(𝜌₁(𝑤_𝑖) − 𝜌₁(𝑤_𝑗))²+ 4𝛾_𝑖𝛾_𝑗 2

}} }} }} } ,

(10) where𝛾_𝑖 = (∑_{𝑘:𝑘∼𝑖}𝜌₁(𝑤_𝑖𝑘)𝜌₁(𝑤_𝑘))/𝜌₁(𝑤_𝑖)and𝑤_𝑖𝑗is the posi- tive definite weight matrix of order p of the edge𝑖𝑗. Moreover equality holds in(10)if and only if

(i)𝐺is a weighted-regular graph or𝐺is a weight-semi- regular bipartite graph;

(ii)𝑤_𝑖𝑗 have a common eigenvector corresponding to the largest eigenvalue𝜌₁(𝑤_𝑖𝑗)for all𝑖,𝑗.

Corollary 9 (see [2]). Let𝐺be a simple connected weighted graph where each edge weight𝑤_𝑖𝑗is a positive number. Then

𝜆₁≤max

𝑖 {𝑤_𝑖+ 𝑤_𝑖} , (11)

where𝑤_𝑖 = (∑_{𝑘:𝑘∼𝑖}𝑤_𝑖𝑘𝑤_𝑘)/𝑤_𝑖and𝑤_𝑖 is the weight of vertex 𝑖. Moreover equality holds if and only if 𝐺 is a bipartite semiregular graph or𝐺is a bipartite regular graph.

Theorem 10 (see [3]). Let G be a simple connected weighted graph. Then

𝜆₁

≤max_𝑖 {{ {{ {{ {

√ 𝜌₁²(𝑤_𝑖) + ∑

𝑘:𝑘∼𝑖𝜌₁²(𝑤_𝑖𝑘) + ∑

𝑘:𝑘∼𝑖𝜌₁(𝑤_𝑖𝑤_𝑖𝑘+ 𝑤_𝑖𝑘𝑤_𝑘) + ∑1≤𝑖,𝑡≤𝑛 ∑

𝑠∈𝑁_𝑖∩𝑁_𝑡𝜌₁(𝑤_𝑖𝑠𝑤_𝑠𝑡)

}} }} }} } ,

(12) where𝑤_𝑖𝑘is the positive definite weight matrix of order𝑝of the edge𝑖𝑘and𝑁_𝑖∩ 𝑁_𝑘is the set of common neighbours of𝑖and 𝑘. Moreover equality holds in(12)if and only if

(i)𝐺is a weight-semiregular bipartite graph;

(ii)𝑤_𝑖𝑘 have a common eigenvector corresponding to the largest eigenvalue𝜌₁(𝑤_𝑖𝑘)for all𝑖,𝑘.

Corollary 11 (see [3]). Let𝐺be a simple connected weighted graph where each edge weight𝑤_𝑖𝑗is a positive number. Then

𝜆₁≤max

𝑖

{{ {{ {{ {

√ 𝑤²_𝑖 + ∑

𝑘:𝑘∼𝑖𝑤_𝑖𝑘² + ∑

𝑘:𝑘∼𝑖(𝑤_𝑖𝑤_𝑖𝑘+ 𝑤_𝑘𝑤_𝑖𝑘) + ∑1≤𝑖,𝑡≤𝑛 ∑

𝑠∈𝑁_𝑖∩𝑁_𝑡𝑤_𝑖𝑠𝑤_𝑠𝑡

}} }} }} }

. (13)

Moreover equality holds if and only if𝐺is a bipartite semireg- ular graph.

3. Two Upper Bounds on the Largest Laplacian Eigenvalue of Weighted Graphs

In this section we present two upper bounds for the largest eigenvalue of weighted graphs and compare the bounds with some examples.

Theorem 12. Let G be a simple connected weighted graph.

Then

𝜆₁≤max

𝑖∼𝑗

{{ {{ {{ {{ {{ {{ {{ {

𝜌₁(𝑤_𝑖) + 𝜌₁(𝑤_𝑗) + √(𝜌₁(𝑤_𝑖) − 𝜌₁(𝑤_𝑗))²+ 4 ( ∑

𝑘:𝑘∼𝑖𝜌₁(𝑤_𝑖𝑘)) ( ∑

𝑘:𝑘∼𝑗𝜌₁(𝑤_𝑗𝑘)) 2

}} }} }} }} }} }} }} }

, (14)

where𝑤_𝑖𝑗is the positive definite weight matrix of order p of the edge𝑖𝑗. Moreover equality holds in(14)if and only if

(i)𝐺 is a weighted-regular graph or 𝐺 is a weight- semiregular bipartite graph;

(ii)𝑤_𝑖𝑗 have a common eigenvector corresponding to the largest eigenvalue𝜌₁(𝑤_𝑖𝑗)for all𝑖,𝑗.

Proof. Let 𝑋 = (𝑥₁, 𝑥₂, . . . , 𝑥_𝑛)^𝑇 be an eigenvector corre- sponding to the largest eigenvalue𝜆₁of𝐿(𝐺). We assume that 𝑥_𝑖is the vector component of𝑋such that

𝑥^𝑇_𝑖𝑥_𝑖=max

𝑘∈𝑉 {𝑥^𝑇_𝑘𝑥_𝑘} . (15)

Since𝑋is nonzero, so is𝑥_𝑖. Let 𝑥^𝑇_𝑗𝑥_𝑗 =max

𝑘:𝑘∼𝑖{𝑥^𝑇_𝑘𝑥_𝑘} (16)

be. The(𝑖, 𝑗)th element of𝐿(𝐺)is {{

{{ {

𝑤_𝑖; if 𝑖 = 𝑗

−𝑤_𝑖,𝑗; if 𝑖 ∼ 𝑗 0; otherwise.

(17)

We have

𝐿 (𝐺) 𝑋 = 𝜆₁𝑋. (18)

From the𝑖th equation of (18), we have (𝜆₁𝐼_𝑝,𝑝− 𝑤_𝑖) 𝑥_𝑖= − ∑

𝑘:𝑘∼𝑖

𝑤_𝑖𝑘𝑥_𝑘, (19)

that is,

𝑥^𝑇_𝑖 (𝜆₁𝐼_𝑝,𝑝− 𝑤_𝑖) 𝑥_𝑖= − ∑

𝑘:𝑘∼𝑖

𝑥^𝑇_𝑖𝑤_𝑖𝑘𝑥_𝑘 (20)

≤ ∑

𝑘:𝑘∼𝑖󵄨󵄨󵄨󵄨󵄨𝑥^𝑇𝑖𝑤_𝑖𝑘𝑥_𝑘󵄨󵄨󵄨󵄨󵄨 (21)

≤ ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘) √𝑥^𝑇_𝑖𝑥_𝑖√𝑥^𝑇_𝑘𝑥_𝑘 by(4) (22)

≤ ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘) √𝑥^𝑇_𝑖𝑥_𝑖√𝑥^𝑇_𝑗𝑥_𝑗 by(16) . (23) From (23) we have

(𝜆₁− 𝜌₁(𝑤_𝑖)) 𝑥^𝑇_𝑖𝑥_𝑖≤ 𝑥^𝑇_𝑖 (𝜆₁𝐼_𝑝,𝑝− 𝑤_𝑖) 𝑥_𝑖 by (2)

≤ ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘) √𝑥^𝑇_𝑖𝑥_𝑖√𝑥^𝑇_𝑗𝑥_𝑗, (24)

that is,

(𝜆₁− 𝜌₁(𝑤_𝑖)) 𝑥^𝑇_𝑖𝑥_𝑖≤ ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘) √𝑥^𝑇_𝑖𝑥_𝑖√𝑥^𝑇_𝑗𝑥_𝑗. (25) From the𝑗th equation of (18), we get

(𝜆₁𝐼_𝑝,𝑝− 𝑤_𝑗) 𝑥_𝑗= − ∑

𝑘:𝑘∼𝑗

𝑤_𝑗𝑘𝑥_𝑘, (26)

that is,

𝑥^𝑇_𝑗 (𝜆₁𝐼_𝑝,𝑝− 𝑤_𝑖) 𝑥_𝑗 = − ∑

𝑘:𝑘∼𝑗

𝑥^𝑇_𝑗𝑤_𝑗𝑘𝑥_𝑘 (27)

≤ ∑

𝑘:𝑘∼𝑗󵄨󵄨󵄨󵄨󵄨𝑥^𝑇𝑗𝑤_𝑗𝑘𝑥_𝑘󵄨󵄨󵄨󵄨󵄨 (28)

≤ ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘) √𝑥^𝑇_𝑗𝑥_𝑗√𝑥^𝑇_𝑘𝑥_𝑘 by(4) (29)

≤ ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘) √𝑥^𝑇_𝑖𝑥_𝑖√𝑥^𝑇_𝑗𝑥_𝑗 by (15) . (30) Similarly, from (30) we get

(𝜆₁− 𝜌₁(𝑤_𝑗)) 𝑥^𝑇_𝑗𝑥_𝑗 ≤ 𝑥^𝑇_𝑗 (𝜆₁𝐼_𝑝,𝑝− 𝑤_𝑗) 𝑥_𝑗 by(2)

≤ ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘) √𝑥^𝑇_𝑖𝑥_𝑖√𝑥^𝑇_𝑗𝑥_𝑗, (31)

that is,

(𝜆₁− 𝜌₁(𝑤𝑗)) 𝑥^𝑇_𝑗𝑥_𝑗 ≤ ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘) √𝑥^𝑇_𝑖𝑥_𝑖√𝑥^𝑇_𝑘𝑥_𝑘. (32)

So, from (25) and (32) we have

(𝜆₁− 𝜌₁(𝑤_𝑗)) (𝜆₁− 𝜌₁(𝑤_𝑖)) 𝑥^𝑇_𝑗𝑥_𝑗𝑥^𝑇_𝑖𝑥_𝑖

≤ ( ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘)) ⋅ ( ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘)) 𝑥^𝑇_𝑗𝑥_𝑗𝑥^𝑇_𝑖𝑥_𝑖. (33)

Hence we get

(𝜆₁− 𝜌₁(𝑤_𝑗)) (𝜆₁− 𝜌₁(𝑤_𝑖))

≤ ( ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘)) ⋅ ( ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘)) , (34)

that is,

𝜆²₁− 𝜆₁(𝜌₁(𝑤_𝑗) + 𝜌₁(𝑤_𝑖)) + 𝜌₁(𝑤_𝑖) 𝜌₁(𝑤_𝑗)

− ( ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘)) ⋅ ( ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘)) ≤ 0, (35)

that is,

𝜆₁ ≤ 𝜌₁(𝑤_𝑖) + 𝜌₁(𝑤_𝑗) 2

+

√(𝜌₁(𝑤_𝑖) − 𝜌₁(𝑤_𝑗))²+ 4 ( ∑

𝑘:𝑘∼𝑖𝜌₁(𝑤_𝑖𝑘)) ( ∑

𝑘:𝑘∼𝑗𝜌₁(𝑤_𝑗𝑘)) 2

≤max

𝑖∼𝑗

{{ {{ {{ {{ {{ {{ {{ {

𝜌₁(𝑤_𝑖)+𝜌₁(𝑤_𝑗) 2

+

√(𝜌₁(𝑤_𝑖)−𝜌₁(𝑤_𝑗))²+4 ( ∑

𝑘:𝑘∼𝑖𝜌₁(𝑤_𝑖𝑘)) ( ∑

𝑘:𝑘∼𝑗𝜌₁(𝑤_𝑗𝑘)) 2

}} }} }} }} }} }} }} } .

(36) This completes the proof of (14).

Now suppose that equality holds in (14). Then all inequal- ities in the previous argument must be equalities.

From equality in (23), we get

𝑥^𝑇_𝑖𝑥_𝑖= 𝑥^𝑇_𝑘𝑥_𝑘 for all𝑘, 𝑘 ∼ 𝑖. (37) Since𝑥_𝑖 ̸= 0, we get that𝑥_𝑘 ̸= 0for all𝑘,𝑘 ∼ 𝑖. From equa- lity in (22) andLemma 2, we get that𝑥_𝑖is an eigenvector of𝑤_𝑖𝑘 for the largest eigenvalue𝜌₁(𝑤_𝑖𝑘). Hence we say that𝑥_𝑘 = 𝑎𝑥_𝑖 for some𝑎, for any𝑘,𝑘 ∼ 𝑖.

On the other hand, from (37) we get

𝑎²𝑥^𝑇_𝑖𝑥_𝑖= 𝑥^𝑇_𝑖𝑥_𝑖, (38) that is,

𝑎²= 1 as𝑥^𝑇_𝑖𝑥_𝑖> 0. (39) From equality in (21), we have

− ∑

𝑘:𝑘∼𝑖

𝑥^𝑇_𝑖𝑤_𝑖𝑘𝑥_𝑘= ∑

𝑘:𝑘∼𝑖󵄨󵄨󵄨󵄨󵄨𝑥^𝑇𝑖𝑤_𝑖𝑘𝑥_𝑘󵄨󵄨󵄨󵄨󵄨 . (40) Since𝑥_𝑘= 𝑎𝑥_𝑖, from (40) we get

− ∑ 𝑎

𝑘:𝑘∼𝑖

𝑥^𝑇_𝑖𝑤_𝑖𝑘𝑥_𝑖= ∑

𝑘:𝑘∼𝑖

|𝑎|󵄨󵄨󵄨󵄨󵄨𝑥^𝑇𝑖𝑤_𝑖𝑘𝑥_𝑘󵄨󵄨󵄨󵄨󵄨

= ∑

𝑘:𝑘∼𝑖

|𝑎| 𝑥^𝑇_𝑖𝑤_𝑖𝑘𝑥_𝑘 as𝑥^𝑇_𝑖𝑤_𝑖𝑘𝑥_𝑘> 0. (41) Hence we get

𝑎 = −1 (42)

from equalities in (41). Therefore we have

𝑥_𝑘 = −𝑥_𝑖 for all𝑘, 𝑘 ∼ 𝑖. (43)

Similarly from equality in (29), we get that 𝑥_𝑗 is an eigenvector of𝑤_𝑗𝑘for the largest eigenvalue𝜌₁(𝑤_𝑗𝑘). Hence we say that𝑥_𝑘 = 𝑏𝑥_𝑗 for some𝑏, for any𝑘,𝑘 ∼ 𝑗. From equality in (16) we have

𝑥^𝑇_𝑗𝑥_𝑗 = 𝑥^𝑇_𝑘𝑥_𝑘 for𝑘 ∼ 𝑖, (44) that is,

𝑏²𝑥^𝑇_𝑗𝑥_𝑗 = 𝑥^𝑇_𝑗𝑥_𝑗, (45) that is,

𝑏²= 1 as𝑥^𝑇_𝑗𝑥_𝑗 > 0. (46) Applying the same methods as previously, we get

𝑏 = −1. (47)

Therefore we have

𝑥_𝑘 = −𝑥_𝑗 for all𝑘, 𝑘 ∼ 𝑗. (48) For𝑖 ∼ 𝑗

𝑥_𝑖= −𝑥_𝑗. (49) Hence we take that𝑈 = {𝑘 : 𝑥_𝑘 = 𝑥_𝑖}and𝑊 = {𝑘 : 𝑥_𝑘=

−𝑥_𝑖}from (43), (48), and (49). So,𝑁_𝑗 ⊂ 𝑈and𝑁_𝑖⊂ 𝑊. Also, 𝑈 ̸= 𝑊 ̸= 0since𝑥_𝑖 ̸= 0. Further, for any vertex𝑠 ∈ 𝑁_𝑁𝑖there exists a vertex𝑟 ∈ 𝑁_𝑖such that𝑟 ∼ 𝑗ℓ𝑟 ∼ 𝑠, where𝑁_𝑁𝑖 is the neighbor of neighbor set of vertex𝑖. Therefore𝑥_𝑟 = −𝑥_𝑖 and𝑥_𝑠= 𝑥_𝑖. So𝑁_𝑁𝑖 ⊂ 𝑈. By similar argument we can present that𝑁_𝑁𝑗 ⊂ 𝑊. Continuing the procedure, it is easy to see, since𝐺is connected, that𝑉 = 𝑈 ∪ 𝑊and that the subgraphs induced by𝑈and𝑊, respectively, are empty graphs. Hence 𝐺is bipartite. Moreover,𝑥_𝑖is a common eigenvector of𝑤_𝑖𝑘 and𝑤_𝑖for the largest eigenvalue𝜌₁(𝑤_𝑖𝑘)and𝜌₁(𝑤_𝑖).

For𝑖, 𝑘 ∈ 𝑈

𝜆₁𝑥_𝑖= 𝑤_𝑖𝑥_𝑖+ ∑

𝑘:𝑘∼𝑖

𝑤_𝑖𝑘𝑥_𝑖= 𝑤_𝑘𝑥_𝑖+ ∑

𝑘:𝑘∼𝑖

𝑤_𝑖𝑘𝑥_𝑖, (50) that is,

𝑤_𝑖𝑥_𝑖= 𝑤_𝑘𝑥_𝑖. (51) Since 𝑥_𝑖 is an eigenvector of 𝑤_𝑖 corresponding to the largest eigenvalue of𝜌₁(𝑤_𝑖)for all𝑖, we get

𝜌₁(𝑤_𝑖) 𝑥_𝑖= 𝜌₁(𝑤_𝑘) 𝑥_𝑖, (52) that is,

(𝜌₁(𝑤_𝑖) − 𝜌₁(𝑤_𝑘)) 𝑥_𝑖= 0, (53) that is,

𝜌₁(𝑤_𝑖) = 𝜌₁(𝑤_𝑘) as𝑥_𝑖 ̸= 0. (54) Therefore we get that 𝜌₁(𝑤_𝑖) is constant for all𝑖 ∈ 𝑈.

Similarly we can show that𝜌₁(𝑤_𝑗)is constant for all𝑗 ∈ 𝑊.

Hence𝐺is a bipartite semiregular graph.

Conversely, suppose that conditions (i)-(ii) of the the- orem hold for the graph 𝐺. Let 𝐺 be (𝜌₁(𝑤_𝑖), 𝜌₁(𝑤_𝑗))- semiregular bipartite graph. Let𝑥be a common eigenvector of𝑤_𝑖𝑘corresponding to the largest eigenvalue𝜌₁(𝑤_𝑖𝑘)for all 𝑖, 𝑘. Then we have

𝜌₁(𝑤_𝑖) = ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘) ,

𝜌₁(𝑤_𝑗) = ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘) . (55)

ByLemma 3, we get

𝜆₁= 𝜌₁(𝑤_𝑖) + 𝜌₁(𝑤_𝑗) , (56) that is,

𝜆₁ =𝜌₁(𝑤_𝑖) + 𝜌₁(𝑤_𝑗) 2

+

√(𝜌₁(𝑤_𝑖) − 𝜌₁(𝑤_𝑗))²+ 4 ( ∑

𝑘:𝑘∼𝑖𝜌₁(𝑤_𝑖𝑘)) ( ∑

𝑘:𝑘∼𝑗𝜌₁(𝑤_𝑗𝑘))

2 .

(57) Corollary 13 (see [1]). Let𝐺be a simple connected weighted graph where each edge weight𝑤_𝑖,𝑗is a positive number. Then

𝜆₁≤max

𝑖∼𝑗 {𝑤_𝑖+ 𝑤_𝑗} . (58)

Moreover equality holds in(58) if and only if𝐺 is bipartite semiregular graph.

Proof. We have𝜌₁(𝑤_𝑖) = 𝑤_𝑖and𝜌₁(𝑤_𝑖𝑗) = 𝑤_𝑖𝑗for all𝑖,𝑗. From Theorem 12, we get the required result.

Corollary 14 (see [5]). Let𝐺be a simple connected unweight- ed graph. Then

𝜆₁≤max

𝑖∼𝑗 {𝑑_𝑖+ 𝑑_𝑗} , (59)

where𝑑_𝑖 is the degree of vertex𝑖. Moreover equality holds in (59) if and only if 𝐺 is a bipartite regular graph or 𝐺 is a bipartite semiregular graph.

Proof. For unweighted graph,𝑤_𝑖,𝑗 = 1for𝑖 ∼ 𝑗. Therefore 𝑤_𝑖= 𝑑_𝑖. UsingCorollary 6, we get the required results.

Theorem 15. Let G be a simple connected weighted graph.

Then 𝜆₁

≤max

𝑖∼𝑗 {√(𝜌₁(𝑤_𝑖) + ∑

𝑘:𝑘∼𝑖𝜌₁(𝑤_𝑖𝑘)) (𝜌₁(𝑤_𝑗) + ∑

𝑘:𝑘∼𝑗𝜌₁(𝑤_𝑗𝑘))} , (60)

where𝑤_𝑖𝑗is the positive definite weight matrix of order p of the edge𝑖𝑗. Moreover equality holds in(60)if and only if

(i)𝐺is a weighted-regular bipartite graph;

(ii)𝑤_𝑖𝑗 have a common eigenvector corresponding to the largest eigenvalue𝜌₁(𝑤_𝑖𝑗)for all𝑖,𝑗.

Proof. Let 𝑋 = (𝑥₁, 𝑥₂, . . . , 𝑥_𝑛)^𝑇 be an eigenvector corre- sponding to the largest eigenvalue𝜆₁of𝐿(𝐺). We assume that 𝑥_𝑖is the vector component of𝑋such that

𝑥^𝑇_𝑖𝑥_𝑖=max

𝑘∈𝑉 {𝑥^𝑇_𝑘𝑥_𝑘} . (61)

Since𝑋is nonzero, so is𝑥_𝑖. Let 𝑥^𝑇_𝑗𝑥_𝑗 =max

𝑘:𝑘∼𝑗{𝑥^𝑇_𝑘𝑥_𝑘} (62)

be. We have

𝐿 (𝐺) 𝑋 = 𝜆₁𝑋. (63) From the𝑖th equation of (43), we have

𝜆₁𝑥_𝑖= 𝑤_𝑖𝑥_𝑖− ∑

𝑘:𝑘∼𝑖

𝑤_𝑖𝑘𝑥_𝑘, (64)

that is,

𝜆₁𝑥^𝑇_𝑖𝑥_𝑖= 󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑥^𝑇_𝑖𝑤_𝑖𝑥_𝑖− ∑

𝑘:𝑘∼𝑖

𝑥^𝑇_𝑖𝑤_𝑖𝑘𝑥_𝑘󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤ 󵄨󵄨󵄨󵄨󵄨𝑥^𝑇𝑖𝑤_𝑖𝑥_𝑖󵄨󵄨󵄨󵄨󵄨 + ∑

𝑘:𝑘∼𝑖󵄨󵄨󵄨󵄨󵄨𝑥^𝑇𝑖𝑤_𝑖𝑘𝑥_𝑘󵄨󵄨󵄨󵄨󵄨

≤ 𝜌₁(𝑤_𝑖) 𝑥^𝑇_𝑖𝑥_𝑖+ ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘) √𝑥^𝑇_𝑖𝑥_𝑖√𝑥^𝑇_𝑘𝑥_𝑘 by(2)

≤ 𝜌₁(𝑤_𝑖) 𝑥^𝑇_𝑖𝑥_𝑖+ ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘) 𝑥^𝑇_𝑖𝑥_𝑖 by (40) . (65) Hence we get

𝜆₁≤ 𝜌₁(𝑤_𝑖) + ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘) . (66) By the same method, from the𝑗th equation of (43), we have

𝜆₁𝑥_𝑗= 𝑤_𝑗𝑥_𝑗− ∑

𝑘:𝑘∼𝑗

𝑤_𝑗𝑘𝑥_𝑘, (67)

that is, 𝜆₁𝑥^𝑇_𝑗𝑥_𝑗 = 󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨𝑥^𝑇_𝑗𝑤_𝑗𝑥_𝑗− ∑

𝑘:𝑘∼𝑗

𝑥^𝑇_𝑗𝑤_𝑗𝑘𝑥_𝑘󵄨󵄨󵄨󵄨

󵄨󵄨󵄨󵄨󵄨󵄨󵄨

≤ 󵄨󵄨󵄨󵄨󵄨𝑥^𝑇𝑗𝑤_𝑗𝑥_𝑗󵄨󵄨󵄨󵄨󵄨 + ∑

𝑘:𝑘∼𝑗󵄨󵄨󵄨󵄨󵄨𝑥^𝑇𝑗𝑤_𝑗𝑘𝑥_𝑘󵄨󵄨󵄨󵄨󵄨

≤ 𝜌₁(𝑤_𝑗) 𝑥^𝑇_𝑗𝑥_𝑗+ ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘) √𝑥^𝑇_𝑗𝑥_𝑗√𝑥^𝑇_𝑘𝑥_𝑘 by(2)

≤ 𝜌₁(𝑤_𝑗) 𝑥^𝑇_𝑗𝑥_𝑗+ ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘) 𝑥^𝑇_𝑗𝑥_𝑗 by(41) . (68) Hence we get

𝜆₁≤ 𝜌₁(𝑤_𝑗) + ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘) . (69)

From (49) and (58), we have

𝜆²₁≤ (𝜌₁(𝑤_𝑖) + ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘)) (𝜌₁(𝑤_𝑗) + ∑

𝑘:𝑘∼𝑗

𝜌₁(𝑤_𝑗𝑘)) , (70) that is,

𝜆₁

≤max

𝑖∼𝑗 {√(𝜌1(𝑤_𝑖) + ∑

𝑘:𝑘∼𝑖𝜌₁(𝑤_𝑖𝑘)) (𝜌₁(𝑤_𝑗) + ∑

𝑘:𝑘∼𝑗𝜌₁(𝑤_𝑗𝑘))} . (71) This completes the proof of (60).

Now we show the case of equality in (60). By similar method inTheorem 12. In the part of equalit, the necessary condition can show easily. So we will show the sufficient condition.

Suppose that conditions (i)-(ii) of Theorem hold for the graph𝐺. We must prove that

𝜆₁

=max

𝑖∼𝑗 {√(𝜌₁(𝑤_𝑖) + ∑

𝑘:𝑘∼𝑖𝜌₁(𝑤_𝑖𝑘)) (𝜌₁(𝑤_𝑗) + ∑

𝑘:𝑘∼𝑗𝜌₁(𝑤_𝑗𝑘))} . (72) Let 𝐺 be regular bipartite graph. Therefore we have 𝜌₁(𝑤_𝑖) = 𝛼 for𝑖 ∈ 𝑈and 𝜌₁(𝑤_𝑗) = 𝛼 for 𝑗 ∈ 𝑊 such that 𝑉 = 𝑈 ∪ 𝑊. Let 𝑥 be a common eigenvector of𝑤_𝑖𝑘 corresponding to the largest eigenvalue 𝜌₁(𝑤_𝑖𝑘) for all 𝑖, 𝑘.

Hence we have

𝜌₁(𝑤_𝑖) = ∑

𝑘:𝑘∼𝑖

𝜌₁(𝑤_𝑖𝑘) . (73)

From (71) we get that

𝜆₁≤ 2𝛼. (74)

On the other hand, the following equation can be easily verified:

(2𝛼) (( (( (( (

( 𝑥𝑥

...

−𝑥𝑥

−𝑥 ...

−𝑥 )) )) )) )

)

=((((

(

𝑤₁ ⋅ 0 −𝑤_1,𝑘+1 ⋅ −𝑤_1,𝑛

0 ⋅ 0 −𝑤_2,𝑘+1 ⋅ −𝑤_2,𝑛

. . . ⋅ . . . ⋅ . . .

0 ⋅ 𝑤_𝑘 −𝑤_𝑘,𝑘+1 ⋅ −𝑤_𝑘,𝑛

−𝑤_𝑘+1,1 ⋅ −𝑤_𝑘+1,𝑘 𝑤_𝑘+1 ⋅ 0

−𝑤_𝑘+1,2 ⋅ −𝑤_𝑘+2,𝑘 0 ⋅ 0

. . . ⋅ . . . ⋅ . . .

−𝑤_𝑛,1 ⋅ −𝑤_𝑛,𝑘 0 ⋅ 𝑤_𝑛 )) ))

)

× (( (( (( (

( 𝑥𝑥

...

−𝑥𝑥

−𝑥 ...

−𝑥 )) )) )) )

) .

(75) Thus2𝛼is an eigenvalue of𝐿(𝐺). Since𝜆₁is the largest eigenvalue of𝐿(𝐺), we get

2𝛼 ≤ 𝜆₁. (76)

So from (74) and (76) we obtain 𝜆₁

=max

𝑖∼𝑗

{{ {

√(𝜌₁(𝑤_𝑖) + ∑

𝑘:𝑘∼𝑖𝜌₁(𝑤_𝑖𝑘)) (𝜌₁(𝑤_𝑗) + ∑

𝑘:𝑘∼𝑗𝜌₁(𝑤_𝑗𝑘))} }} . (77) Corollary 16. Let 𝐺 be a simple connected weighted graph where each edge weight𝑤_𝑖,𝑗is a positive number. Then

𝜆₁≤max

𝑖∼𝑗 {2√𝑤𝑖𝑤_𝑗} . (78)

Moreover equality holds in(78) if and only if𝐺 is bipartite semiregular graph.

Proof. We have𝜌₁(𝑤_𝑖) = 𝑤_𝑖and𝜌₁(𝑤_𝑖𝑗) = 𝑤_𝑖𝑗for all𝑖,𝑗. From Theorem 15we get the required result.

Corollary 17. Let𝐺be a simple connected unweighted graph.

Then

𝜆₁≤max

𝑖∼𝑗 {2√𝑑_𝑖𝑑_𝑗} , (79)

where𝑑_𝑖 is the degree of vertex𝑖. Moreover equality holds in (79) if and only if 𝐺 is a bipartite regular graph or 𝐺 is a bipartite semiregular graph.

Proof. For unweighted graph,𝑤_𝑖,𝑗 = 1for𝑖 ∼ 𝑗. Therefore 𝑤_𝑖= 𝑑_𝑖. UsingCorollary 16, we get the required results.

Example 18. Let 𝐺₁ = (𝑉₁, 𝐸₁) and 𝐺₂ = (𝑉₂, 𝐸₂) be a weighted graph where𝑉₁ = {1, 2, 3, 4},𝐸₁ = {{1, 3}, {2, 4}, {3, 4}}and each weight is the positive definite matrix of order three. Let𝑉₂ = {1, 2, 3, 4, 5, 6, 7},𝐸₂ = {{1,4},{2,4},{3,4},

{4,5},{5,6},{5,7}}such that each weight is the positive definite matrix of order two.

Assume that the following Laplacian matrices of𝐺₁and𝐺₂ are as follows:

𝐿 (𝐺₁) = [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [ [

1 0 0 0 0 0 −1 0 0 0 0 0 0 5 3 0 0 0 0 −5 −3 0 0 0 0 3 3 0 0 0 0 −3 −3 0 0 0 0 0 0 2 −1 0 0 0 0 −2 1 0 0 0 0 −1 2 −1 0 0 0 1 −2 1 0 0 0 0 −1 2 0 0 0 0 1 −2

−1 0 0 0 0 0 7 2 −2 6 2 −2 0 −5 −3 0 0 0 2 11 1 2 6 −2 0 −3 −3 0 0 0 −2 1 13 −2 −2 10 0 0 0 2 −1 0 6 2 −2 8 1 −2 0 0 0 −1 2 −1 2 6 −2 −3 8 −3 0 0 0 0 −1 2 −2 −2 10 −2 −2 12

]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ] ] ,

𝐿 (𝐺₂) = [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [[ [

1 1 0 0 0 0 −1 −1 0 0 0 0 0 0 1 2 0 0 0 0 −1 −2 0 0 0 0 0 0 0 0 1 1 0 0 −1 −1 0 0 0 0 0 0 0 0 1 3 0 0 −1 −3 0 0 0 0 0 0 0 0 0 0 1 1 −1 −1 0 0 0 0 0 0 0 0 0 0 1 4 −1 −4 0 0 0 0 0 0

−1 −1 −1 −1 −1 −1 4 4 −1 −1 0 0 0 0

−1 −2 −1 −3 −1 −4 4 14 −1 −5 0 0 0 0 0 0 0 0 0 0 −1 −1 3 3 −1 −1 −1 −1 0 0 0 0 0 0 −1 −5 3 18 −1 −6 −1 7 0 0 0 0 0 0 0 0 −1 −1 1 1 0 0 0 0 0 0 0 0 0 0 −1 −6 1 6 0 0 0 0 0 0 0 0 0 0 −1 −1 0 0 1 1 0 0 0 0 0 0 0 0 −1 −7 0 0 1 7

]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ]] ] .

(80) The largest eigenvalues of𝐿(𝐺₁)and𝐿(𝐺₂)are𝜆₁ = 25, 66,

𝜆₂ = 26.16rounded two decimal places and the previously mentioned bounds give the following results:

(6) (7) (10) (12) (14) (60) 𝐺₁ 32.90 32.88 27.90 29.55 30.88 30.93

𝐺₂ 34.12 29.86 27.11 27.22 34.05 33.90. (81) For𝐺₁, we see that the upper bounds in (14) and (60) are better than upper bounds in (6) and (7). But they are not better than upper bounds in (10) and (12) from (81).

For also𝐺₂, we see that upper bounds in (14) and (60) are only better than the upper bound in (6).

Consequently, we cannot exactly compare all the bounds for weighted graphs, where the weights are positive definite matrices. Modifications according to each weight of edges, especially for matrices can be shown.

References

[1] K. Ch. Das and R. B. Bapat, “A sharp upper bound on the largest Laplacian eigenvalue of weighted graphs,”Linear Algebra and Its Applications, vol. 409, pp. 153–165, 2005.

[2] K. Ch. Das, “Extremal graph characterization from the upper bound of the Laplacian spectral radius of weighted graphs,”

Linear Algebra and Its Applications, vol. 427, no. 1, pp. 55–69, 2007.

[3] S. Sorgun and S¸. B¨uy¨ukk¨ose, “On the bounds for the largest Lap- lacian eigenvalues of weighted graphs,”Discrete Optimization, vol. 9, no. 2, pp. 122–129, 2012.

[4] R. A. Horn and C. R. Johnson, Matrix Analysis, Cambridge University Press, Cambridge, UK, 1985.

[5] W. N. Anderson Jr. and T. D. Morley, “Eigenvalues of the Lapla- cian of a graph,”Linear and Multilinear Algebra, vol. 18, no. 2, pp. 141–145, 1985.

Submit your manuscripts at http://www.hindawi.com

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation http://www.hindawi.com

Differential Equations

International Journal of

Volume 2014

Applied Mathematics^{Journal of}

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Mathematical PhysicsAdvances in

Complex Analysis

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Optimization

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Combinatorics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

International Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Journal of

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Function Spaces

Abstract and Applied Analysis

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

The Scientific World Journal

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Discrete Mathematics

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Hindawi Publishing Corporation

http://www.hindawi.com Volume 2014

Stochastic Analysis

International Journal of