Research Article Sharp Upper Bounds for the Laplacian ...downloads.hindawi.com/journals/mpe/2013/720854.pdf · the Laplacian spectral radius of irregulargraphs as follows. Let be

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 720854 4 pageshttpdxdoiorg1011552013720854

Research ArticleSharp Upper Bounds for the Laplacian SpectralRadius of Graphs

Houqing Zhou1 and Youzhuan Xu2

1 Department of Mathematics Shaoyang University Hunan 422000 China2 Shaoyang Radio amp TV University Hunan 422000 China

Correspondence should be addressed to Houqing Zhou zhouhq2004163com

Received 15 August 2013 Accepted 10 October 2013

Academic Editor Miguel A F Sanjuan

Copyright copy 2013 H Zhou and Y Xu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The spectrum of the Laplacian matrix of a network plays a key role in a wide range of dynamical problems associated with thenetwork from transient stability analysis of power network to distributed control of formations Let 119866 = (119881 119864) be a simpleconnected graph on 119899 vertices and let 120583(119866) be the largest Laplacian eigenvalue (ie the spectral radius) of 119866 In this paper byusing the Cauchy-Schwarz inequality we show that the upper bounds for the Laplacian spectral radius of 119866

1 Introduction

Theeigenvalue spectrumof the Laplacianmatrix of a networkprovides valuable information regarding the behavior ofmanydynamical processes taking place on the network In [1]Pecora and Carroll related the problem of synchronization ina network of coupled oscillators to the largest and second-smallest Laplacian eigenvalues (usually denoted by Laplacianspectral radius and spectral gap resp) of the network Morerecently Dorfler and Bullo (see [2]) derived conditions fortransient stability in power networks in terms of the spectralgap of the Laplacian matrix Apart from their applicabilityto the problems of synchronization and transient stabilityanalysis the Laplacian eigenvalues are also relevant in theanalysis of many distributed estimation and control problems(see [3])

Understanding the relationship between the structure ofa complex network and the behavior of dynamical processestaking place in it is a central question in the research fieldof network science Since the behavior of many networkeddynamical processes is closely related to the Laplacian eigen-values it is of interest to study the relationship between struc-tural features of the network and its Laplacian eigenvaluesIn this paper we mainly study the spectral radius of theLaplacian matrix

2 Preliminaries

Let119866 = (119881 119864) be a simple undirected graph on 119899 verticesTheLaplacianmatrix of119866 is the 119899times119899matrix 119871(119866) = 119863(119866)minus119860(119866)where 119860(119866) is the adjacency and119863(119866) is the diagonal matrixof vertex degrees It is well known that 119871(119866) is a positivesemidefinite matrix and that (0 e) is an eigenpair of 119871(119866)where e is the all-ones vector In [4] some of themany resultsknown for Laplacian matrices are given The spectrum of119871(119866) is 120583

1(119866) 120583

2(119866) 120583

119899(119866) where 120583

1(119866) ge 120583

2(119866) ge

sdot sdot sdot ge 120583119899(119866) = 0 The largest eigenvalue 120583

1(119866) is called the

Laplacian spectral radius of the graph 119866 denoted by 120583(119866)For a star graph of order 119899 the spectrum is 119899 1 1 1⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟

119899minus2

0

We recall that upper bounds of the spectral radius of119871(119866)It is a well-known fact that 120583(119866) le 2Δ with equality if andonly if 119866 is bipartite regular Shi [5] gave an upper bound forthe Laplacian spectral radius of irregular graphs as follows

Let 119866 be a connected irregular graph of order 119899 withmaximum degree Δ and diameter 119863 Then 120583(119866) lt 2Δ minus

2(2119863 + 1)119899Li et al [6] improve Shirsquos upper bound for the Laplacian

spectral radius of irregular graphs They show the followingresult

120583 (119866) lt 2Δ minus1

119899119863 (1)

2 Mathematical Problems in Engineering

Dyilek Maden and Buyukkose [7] proved the followingLet 119866 be a simple graph Then

radic119898 +119904

radic119899 minus 1le 1205831(119866) le radic119898 + 119904radic119899 minus 1 (2)

where119898 = (sum119899

119894=1119889119894(119889119894+ 1))119899 and

1199042=

sum119899

119894=1(1198892

119894+ 119889119894)2

119899

+

2sum119894lt119895 119894sim119895

(119889119894+ 119889119895) (119889119894+ 119889119895minus 2

10038161003816100381610038161003816119873119894cap 119873119895

10038161003816100381610038161003816)

119899

+

2sum119894lt119895

10038161003816100381610038161003816119873119894cap 119873119895

10038161003816100381610038161003816

2

119899minus 1198982

(3)

In this paper we continue to consider the upper boundsfor the Laplacian spectral radius of graphs The rest ofthe paper is organized as follows Section 3 contains somelemmas which play a fundamental role Section 4 containstwo theorems on the upper bounds of 120583(119866)

3 Some Useful Lemmas

In the proof of several theorems we will use the followinglemmas

Lemma 1 (see [8]) Let 119866 be a connected graph with 119899 verticesand119898 edges then

119899

sum

119894=1

1198892

119894le2119898 (2119898 + (119899 minus 1) (Δ minus 120575))

119899 + Δ minus 120575 (4)

Equality holds if and only if 119866 = 119861119899119905

for some 1 le 119905 le 119899 or 119866 isregular where119861

119899119905denotes the graph on 119899 vertices with exactly 119905

vertices of degree 119899minus1 and the remaining of 119899minus119905 vertices formingan independent set Notice that 119861

1198991= 1198701119899minus1

and 119861119899119899

= 119870119899

Let119872 be an119898times 119899matrix Then 119904119894(119872) will denote the 119894th

row sum of119872 that is 119904119894(119872) = sum

119899

119895=1119872119894119895 where 1 le 119894 le 119898

Lemma 2 (see [9]) Let 119866 be a connected 119899-vertex graph and119860 its adjacency matrix with spectral radius 120588(119860) Let 119875 be anypolynomial Then

min119894isin119881(119866)

119904119894(119875 (119860)) le 119875 (120588 (119860)) le max

119894isin119881(119866)

119904119894(119875 (119860)) (5)

Lemma 3 (see [10]) Let 119866 be a simple graph with vertex set119881(119866) = V

1 V2 V

119899 Let 120588 denote the spectral radius of the

line graph 119897(119866) of 119866 Then the inequality

120583 (119866) le 2 + 120588 (6)

holds and the equality occurs if and only if 119866 is a bipartitegraph

4 Main Results

In this section we consider simple connected graph with119899 vertices The main result of the paper is the followingtheorem

Theorem4 Let119866 be a graph with 119899 vertices and119898 edges then

120583 (119866) le1

119899 minus 1(2119898 + ( (119899 minus 2)119898 (119899 (119899 minus 1) minus 2119898)

times2 (Δ minus 120575 + 1)

119899 + Δ minus 120575)

12

)

(7)

with equality if and only if 119866 is the star 1198701119899minus1

or the completegraph 119870

119899

Proof Since119899

sum

119894=1

120583119894(119866) = tr (119871 (119866)) = 2119898 (8)

where tr(119871(119866)) denotes the trace of 119871(119866) Notice that1205832

1(119866) 120583

2

2(119866) 120583

2

119899(119866) are eigenvalues of 1198712(119866) hence we

have119899

sum

119894=1

1205832

119894(119866) = tr (1198712 (119866)) = 2119898 +

119899

sum

119894=1

1198892

119894 (9)

According to the Cauchy-Schwarz inequality we have

(119899 minus 2) (1205832

2(119866) + 120583

2

3(119866) + sdot sdot sdot + 120583

2

119899minus1(119866))

ge (1205832(119866) + 120583

3(119866) + sdot sdot sdot + 120583

119899minus1(119866))2

(10)

That is

(119899 minus 2)(2119898 minus 1205832(119866) +

119899

sum

119894=1

1198892

119894) ge (2119898 minus 120583 (119866))

2

(11)

By means of Lemma 1 we obtain

(2119898 minus 120583 (119866))2

le (119899 minus 2) (2119898 minus 120583 (119866)2)

+ (119899 minus 2) sdot2119898 (2119898 + (119899 minus 1) (Δ minus 120575))

119899 + Δ minus 120575

(12)

Suppose that 119905 = (119899minus2) sdot (2119898(2119898+(119899minus1)(Δminus120575)))(119899+Δminus120575)simplifying the inequality above we get

(119899 minus 1) 120583(119866)2minus 4119898120583 (119866) + 4119898

2minus 2119898 (119899 minus 2) minus 119905 le 0 (13)

Hence we have2119898 minus 119908

119899 minus 1le 120583 (119866) le

2119898 + 119908

119899 minus 1 (14)

where 119908 = radic(119899 minus 1)119905 + 2119898(119899 minus 1)(119899 minus 2) minus 41198982(119899 minus 2)Let 119866 = 119870

1119899minus1and then 119898 = 119899 minus 1 119905 = 119899(119899 minus 1)(119899 minus 2)

119908 = (119899 minus 1)(119899 minus 2) we have

2119898 + 119908

119899 minus 1=2 (119899 minus 1) + (119899 minus 1) (119899 minus 2)

119899 minus 1= 119899 = 120583 (119866) (15)

Mathematical Problems in Engineering 3

6 1

2

34

5

Figure 1 The graph 119866 on 6 vertices

Next we examine the complete graph 119870119899 The complete

graph has 119899 vertices and 119899(119899 minus 1)2 edges Δ = 119899 minus 1 = 120575Thus 119905 = 119899(119899 minus 1)2(119899 minus 2) hence 119908 = 0 we get

2119898 + 119908

119899 minus 1= 119899 = 120583 (119866) (16)

Equality holds on the right in (14) if and only if 119866 is thestar 119870

1119899minus1or the complete graph119870

119899

This completes the proof of the theorem

Example 5 In this example we illustrate the technique ofTheorem 4 Consider the graph 119866 on 6 vertices and 8 edgesin Figure 1 this graph has the largest degree Δ = 3 and thesmallest degree 120575 = 2

Now we estimate the largest eigenvalue 120583(119866) withTheorem 4 Applying this upper bound on 120583(119866) it followsthat

0 le 120583 (119866) le 64 (17)

By a straightforward calculation we show that the Laplacianeigenvalues of 119866 are

1205831(119866) = 3 + radic3 120583

2(119866) = 4 120583

3(119866) = 4

1205834(119866) = 2 120583

5(119866) = 3 minus radic3 120583

6(119866) = 0

(18)

Clearly 1205831(119866) = 3 + radic3 lt 64 holds

It is easy to see that we can use themethod to estimate theupper bound of the largest Laplacian eigenvalue

The following Theorem 6 is associated with edge anddegree of graph 119866 that is associated with the largest andthe second largest degree Δ

1015840 the smallest degree of 119866respectively

Theorem 6 Let 119866 = (119881 119864) be a simple graph with 119899 verticesand119898 edges then

120583 (119866) le (Δ1015840+ 120575 minus 1

+ ((Δ1015840+ 120575 minus 1)

+4 (Δ2minus Δ sdot Δ

1015840+ 4119898 minus 2 (119899 minus 1) 120575))

12

)

times (2)minus1

(19)

with equality if and only if 119866 is a regular bipartite graph

Proof Let119860 = (119886119894119895) be the adjacencymatrix of a graph119866with

vertices 1 2 119899 let 119860119896 = (119886(119896)

119894119895) further let119873

119894(119860119896) denote

the number of walks of length 119896 starting at vertex 119894 Hence119873119894(1198602) = sum

119895sim119894119889119895

Similarly for the adjacency matrix 119861 of line graph 119897(119866) of119866 let1198731015840

119906(119861119896) denote the number of walks of length 119896 starting

at vertex 119906 and let 119889119897(119866)

(119894119895) denote the degree of the vertex 119894119895in line graph 119897(119866) then we have

1198731015840

119894119895(119861) = 119889

119897(119866)(119894119895) = 119889

119894+ 119889119895minus 2 (20)

It can easily be seen that

1198731015840

119894119895(1198612) = sum

119901119902sim119894119895

119889119897(119866)

(119901119902) = sum

119901119902sim119894119895

(119889119901+ 119889119902minus 2) (21)

that is

sum

119902sim119894

(119889119902+ 119889119894minus 2) + sum

119901sim119895

(119889119901+ 119889119895minus 2) minus 2 (119889

119894+ 119889119895minus 2)

= 1198892

119894minus 2119889119894+sum

119902sim119894

119889119902+ 1198892

119895minus 2119889119895

+ sum

119901sim119895

119889119901minus 2 (119889

119894+ 119889119895minus 2)

le Δ119889119894+ Δ1015840119889119895+ 4119898 minus 3 (119889

119894+ 119889119895)

minussum

119902≁119894

119889119902minus sum

119901≁119895

119889119901minus 2 (119889

119894+ 119889119895minus 2)

le Δ1015840(119889119894+ 119889119895) + (Δ minus Δ

1015840) Δ + 4119898 minus 3 (119889

119894+ 119889119895)

minus (119899 minus 119889119894minus 1) 120575 minus (119899 minus 119889

119895minus 1) 120575 minus 2 (119889

119894+ 119889119895minus 2)

= (Δ1015840+ 120575 minus 5) (119889

119894+ 119889119895minus 2) + Δ

2

+ (2 minus Δ) Δ1015840minus 2 (119899 minus 2) 120575 + 4119898 minus 6

(22)

According to (20) we have

1198731015840

119894119895((1198612minus (Δ1015840+ 120575 minus 5) 119861))

le Δ2+ (2 minus Δ) Δ

1015840minus 2 (119899 minus 2) 120575 + 4119898 minus 6

(23)

Using Lemma 2 we obtain

1205882minus (Δ1015840+ 120575 minus 5) 120588



(24)


From the inequality above we have

120588 le (Δ1015840+ 120575 minus 5

+ ((Δ1015840+ 120575 minus 5)

2

+4 (Δ2+ (2 minus Δ) Δ

1015840minus 2 (119899 minus 2) 120575 + 4119898 minus 6))

12

)

times (2)minus1

= (Δ1015840+ 120575 minus 5

+ ((Δ1015840+ 120575 minus 1)

2


1015840minus 2 (119899 minus 1) 120575 + 4119898))

12

)

times (2)minus1

(25)

Using Lemma 3 we have

120583 (119866) le (Δ1015840+ 120575 minus 1

+ ((Δ1015840+ 120575 minus 1)


1015840+ 4119898 minus 2 (119899 minus 1) 120575))

12

)

times (2)minus1

(26)

If equality holds in (19) applying Lemma 3 hence wededuce that 119866 is a bipartite graph And by Lemma 2 for any119894119895 isin 119864(119866) the equality in (25) occurs That is for any 119894119895 theequality in (22) holds Then we have

Δ = Δ1015840= 120575 (27)

that is 119866 is a regular graph hence 119866 is a regular bipartiteMore generally if 119866 is a regular bipartite graph then by

verifying straightforward (19) the equality holds

Acknowledgments

The authors thank Professor Miguel A F Sanjuan for hisuseful comments and suggestions Project is supported byHunan Provincial Natural Science Foundation of China no13JJ3118 and by Scientific Research Fund of Shaoyang Scienceamp Technology no M230

References

[1] L M Pecora and T L Carroll ldquoMaster stability functions forsynchronized coupled systemsrdquo Physical Review Letters vol 80no 10 pp 2109ndash2112 1998

[2] F Dorfler and F Bullo ldquoSynchronization and transient stabilityin power networks and non-uniform kuramoto oscillatorsrdquoSIAM Journal on Control and Optimization vol 50 pp 1616ndash1642 2012

[3] R Olfati-Saber J A Fax and R M Murray ldquoConsensus andcooperation in networked multi-agent systemsrdquo Proceedings ofthe IEEE vol 95 no 1 pp 215ndash233 2007

[4] R Merris ldquoLaplacian matrices of graphs a surveyrdquo LinearAlgebra and Its Applications vol 197-198 pp 143ndash176 1994

[5] L Shi ldquoBounds on the (Laplacian) spectral radius of graphsrdquoLinear Algebra and Its Applications vol 422 no 2-3 pp 755ndash770 2007

[6] J LiW C Shiu andA Chang ldquoTheLaplacian spectral radius ofgraphsrdquo Czechoslovak Mathematical Journal vol 60 no 3 pp835ndash847 2010

[7] A Dyilek Maden and S Buyukkose ldquoBounds for Laplaciangraph eigenvaluesrdquo Mathematical Inequalities amp Applicationsvol 12 pp 529ndash536 2012

[8] S M Cioaba ldquoSums of powers of the degrees of a graphrdquoDiscrete Mathematics vol 306 no 16 pp 1959ndash1964 2006

[9] M N Ellingham and X Zha ldquoThe spectral radius of graphs onsurfacesrdquo Journal of Combinatorial Theory B vol 78 no 1 pp45ndash56 2000

[10] J Shu YHong andRWen ldquoA sharp upper boundon the largesteigenvalue of the Laplacian matrix of a graphrdquo Linear Algebraand Its Applications vol 347 no 1ndash3 pp 123ndash129 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Dyilek Maden and Buyukkose [7] proved the followingLet 119866 be a simple graph Then

radic119898 +119904

radic119899 minus 1le 1205831(119866) le radic119898 + 119904radic119899 minus 1 (2)

where119898 = (sum119899

119894=1119889119894(119889119894+ 1))119899 and

1199042=

sum119899

119894=1(1198892

119894+ 119889119894)2

119899

+

2sum119894lt119895 119894sim119895

(119889119894+ 119889119895) (119889119894+ 119889119895minus 2

10038161003816100381610038161003816119873119894cap 119873119895

10038161003816100381610038161003816)

119899

+

2sum119894lt119895

10038161003816100381610038161003816119873119894cap 119873119895

10038161003816100381610038161003816

2

119899minus 1198982

(3)

In this paper we continue to consider the upper boundsfor the Laplacian spectral radius of graphs The rest ofthe paper is organized as follows Section 3 contains somelemmas which play a fundamental role Section 4 containstwo theorems on the upper bounds of 120583(119866)

3 Some Useful Lemmas

In the proof of several theorems we will use the followinglemmas

Lemma 1 (see [8]) Let 119866 be a connected graph with 119899 verticesand119898 edges then

119899

sum

119894=1

1198892

119894le2119898 (2119898 + (119899 minus 1) (Δ minus 120575))

119899 + Δ minus 120575 (4)

Equality holds if and only if 119866 = 119861119899119905

for some 1 le 119905 le 119899 or 119866 isregular where119861

119899119905denotes the graph on 119899 vertices with exactly 119905

vertices of degree 119899minus1 and the remaining of 119899minus119905 vertices formingan independent set Notice that 119861

1198991= 1198701119899minus1

and 119861119899119899

= 119870119899

Let119872 be an119898times 119899matrix Then 119904119894(119872) will denote the 119894th

row sum of119872 that is 119904119894(119872) = sum

119899

119895=1119872119894119895 where 1 le 119894 le 119898

Lemma 2 (see [9]) Let 119866 be a connected 119899-vertex graph and119860 its adjacency matrix with spectral radius 120588(119860) Let 119875 be anypolynomial Then

min119894isin119881(119866)

119904119894(119875 (119860)) le 119875 (120588 (119860)) le max

119894isin119881(119866)

119904119894(119875 (119860)) (5)

Lemma 3 (see [10]) Let 119866 be a simple graph with vertex set119881(119866) = V

1 V2 V

119899 Let 120588 denote the spectral radius of the

line graph 119897(119866) of 119866 Then the inequality

120583 (119866) le 2 + 120588 (6)

holds and the equality occurs if and only if 119866 is a bipartitegraph

4 Main Results

In this section we consider simple connected graph with119899 vertices The main result of the paper is the followingtheorem

Theorem4 Let119866 be a graph with 119899 vertices and119898 edges then

120583 (119866) le1

119899 minus 1(2119898 + ( (119899 minus 2)119898 (119899 (119899 minus 1) minus 2119898)

times2 (Δ minus 120575 + 1)

119899 + Δ minus 120575)

12

)

(7)

with equality if and only if 119866 is the star 1198701119899minus1

or the completegraph 119870

119899

Proof Since119899

sum

119894=1

120583119894(119866) = tr (119871 (119866)) = 2119898 (8)

where tr(119871(119866)) denotes the trace of 119871(119866) Notice that1205832

1(119866) 120583

2

2(119866) 120583

2

119899(119866) are eigenvalues of 1198712(119866) hence we

have119899

sum

119894=1

1205832

119894(119866) = tr (1198712 (119866)) = 2119898 +

119899

sum

119894=1

1198892

119894 (9)

According to the Cauchy-Schwarz inequality we have

(119899 minus 2) (1205832

2(119866) + 120583

2

3(119866) + sdot sdot sdot + 120583

2

119899minus1(119866))

ge (1205832(119866) + 120583

3(119866) + sdot sdot sdot + 120583

119899minus1(119866))2

(10)

That is

(119899 minus 2)(2119898 minus 1205832(119866) +

119899

sum

119894=1

1198892

119894) ge (2119898 minus 120583 (119866))

2

(11)

By means of Lemma 1 we obtain

(2119898 minus 120583 (119866))2

le (119899 minus 2) (2119898 minus 120583 (119866)2)

+ (119899 minus 2) sdot2119898 (2119898 + (119899 minus 1) (Δ minus 120575))

119899 + Δ minus 120575

(12)

Suppose that 119905 = (119899minus2) sdot (2119898(2119898+(119899minus1)(Δminus120575)))(119899+Δminus120575)simplifying the inequality above we get

(119899 minus 1) 120583(119866)2minus 4119898120583 (119866) + 4119898

2minus 2119898 (119899 minus 2) minus 119905 le 0 (13)

Hence we have2119898 minus 119908

119899 minus 1le 120583 (119866) le

2119898 + 119908

119899 minus 1 (14)

where 119908 = radic(119899 minus 1)119905 + 2119898(119899 minus 1)(119899 minus 2) minus 41198982(119899 minus 2)Let 119866 = 119870

1119899minus1and then 119898 = 119899 minus 1 119905 = 119899(119899 minus 1)(119899 minus 2)

119908 = (119899 minus 1)(119899 minus 2) we have

2119898 + 119908

119899 minus 1=2 (119899 minus 1) + (119899 minus 1) (119899 minus 2)

119899 minus 1= 119899 = 120583 (119866) (15)


6 1

2

34

5




2119898 + 119908

119899 minus 1= 119899 = 120583 (119866) (16)



119899




0 le 120583 (119866) le 64 (17)


1205831(119866) = 3 + radic3 120583

2(119866) = 4 120583

3(119866) = 4

1205834(119866) = 2 120583

5(119866) = 3 minus radic3 120583

6(119866) = 0

(18)






120583 (119866) le (Δ1015840+ 120575 minus 1

+ ((Δ1015840+ 120575 minus 1)


1015840+ 4119898 minus 2 (119899 minus 1) 120575))

12

)

times (2)minus1

(19)



vertices 1 2 119899 let 119860119896 = (119886(119896)

119894119895) further let119873

119894(119860119896) denote


119895sim119894119889119895



at vertex 119906 and let 119889119897(119866)


1198731015840

119894119895(119861) = 119889

119897(119866)(119894119895) = 119889

119894+ 119889119895minus 2 (20)


1198731015840

119894119895(1198612) = sum

119901119902sim119894119895

119889119897(119866)

(119901119902) = sum

119901119902sim119894119895

(119889119901+ 119889119902minus 2) (21)

that is

sum

119902sim119894

(119889119902+ 119889119894minus 2) + sum

119901sim119895

(119889119901+ 119889119895minus 2) minus 2 (119889

119894+ 119889119895minus 2)

= 1198892

119894minus 2119889119894+sum

119902sim119894

119889119902+ 1198892

119895minus 2119889119895

+ sum

119901sim119895

119889119901minus 2 (119889

119894+ 119889119895minus 2)

le Δ119889119894+ Δ1015840119889119895+ 4119898 minus 3 (119889

119894+ 119889119895)

minussum

119902≁119894

119889119902minus sum

119901≁119895

119889119901minus 2 (119889

119894+ 119889119895minus 2)

le Δ1015840(119889119894+ 119889119895) + (Δ minus Δ

1015840) Δ + 4119898 minus 3 (119889

119894+ 119889119895)


119895minus 1) 120575 minus 2 (119889

119894+ 119889119895minus 2)

= (Δ1015840+ 120575 minus 5) (119889

119894+ 119889119895minus 2) + Δ

2


(22)


1198731015840

119894119895((1198612minus (Δ1015840+ 120575 minus 5) 119861))



(23)


1205882minus (Δ1015840+ 120575 minus 5) 120588



(24)



120588 le (Δ1015840+ 120575 minus 5

+ ((Δ1015840+ 120575 minus 5)

2

+4 (Δ2+ (2 minus Δ) Δ


12

)

times (2)minus1

= (Δ1015840+ 120575 minus 5

+ ((Δ1015840+ 120575 minus 1)

2


1015840minus 2 (119899 minus 1) 120575 + 4119898))

12

)

times (2)minus1

(25)


120583 (119866) le (Δ1015840+ 120575 minus 1

+ ((Δ1015840+ 120575 minus 1)


1015840+ 4119898 minus 2 (119899 minus 1) 120575))

12

)

times (2)minus1

(26)


Δ = Δ1015840= 120575 (27)



Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










6 1

2

34

5




2119898 + 119908

119899 minus 1= 119899 = 120583 (119866) (16)



119899




0 le 120583 (119866) le 64 (17)


1205831(119866) = 3 + radic3 120583

2(119866) = 4 120583

3(119866) = 4

1205834(119866) = 2 120583

5(119866) = 3 minus radic3 120583

6(119866) = 0

(18)






120583 (119866) le (Δ1015840+ 120575 minus 1

+ ((Δ1015840+ 120575 minus 1)


1015840+ 4119898 minus 2 (119899 minus 1) 120575))

12

)

times (2)minus1

(19)



vertices 1 2 119899 let 119860119896 = (119886(119896)

119894119895) further let119873

119894(119860119896) denote


119895sim119894119889119895



at vertex 119906 and let 119889119897(119866)


1198731015840

119894119895(119861) = 119889

119897(119866)(119894119895) = 119889

119894+ 119889119895minus 2 (20)


1198731015840

119894119895(1198612) = sum

119901119902sim119894119895

119889119897(119866)

(119901119902) = sum

119901119902sim119894119895

(119889119901+ 119889119902minus 2) (21)

that is

sum

119902sim119894

(119889119902+ 119889119894minus 2) + sum

119901sim119895

(119889119901+ 119889119895minus 2) minus 2 (119889

119894+ 119889119895minus 2)

= 1198892

119894minus 2119889119894+sum

119902sim119894

119889119902+ 1198892

119895minus 2119889119895

+ sum

119901sim119895

119889119901minus 2 (119889

119894+ 119889119895minus 2)

le Δ119889119894+ Δ1015840119889119895+ 4119898 minus 3 (119889

119894+ 119889119895)

minussum

119902≁119894

119889119902minus sum

119901≁119895

119889119901minus 2 (119889

119894+ 119889119895minus 2)

le Δ1015840(119889119894+ 119889119895) + (Δ minus Δ

1015840) Δ + 4119898 minus 3 (119889

119894+ 119889119895)


119895minus 1) 120575 minus 2 (119889

119894+ 119889119895minus 2)

= (Δ1015840+ 120575 minus 5) (119889

119894+ 119889119895minus 2) + Δ

2


(22)


1198731015840

119894119895((1198612minus (Δ1015840+ 120575 minus 5) 119861))



(23)


1205882minus (Δ1015840+ 120575 minus 5) 120588



(24)



120588 le (Δ1015840+ 120575 minus 5

+ ((Δ1015840+ 120575 minus 5)

2

+4 (Δ2+ (2 minus Δ) Δ


12

)

times (2)minus1

= (Δ1015840+ 120575 minus 5

+ ((Δ1015840+ 120575 minus 1)

2


1015840minus 2 (119899 minus 1) 120575 + 4119898))

12

)

times (2)minus1

(25)


120583 (119866) le (Δ1015840+ 120575 minus 1

+ ((Δ1015840+ 120575 minus 1)


1015840+ 4119898 minus 2 (119899 minus 1) 120575))

12

)

times (2)minus1

(26)


Δ = Δ1015840= 120575 (27)



Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120588 le (Δ1015840+ 120575 minus 5

+ ((Δ1015840+ 120575 minus 5)

2

+4 (Δ2+ (2 minus Δ) Δ


12

)

times (2)minus1

= (Δ1015840+ 120575 minus 5

+ ((Δ1015840+ 120575 minus 1)

2


1015840minus 2 (119899 minus 1) 120575 + 4119898))

12

)

times (2)minus1

(25)


120583 (119866) le (Δ1015840+ 120575 minus 1

+ ((Δ1015840+ 120575 minus 1)


1015840+ 4119898 minus 2 (119899 minus 1) 120575))

12

)

times (2)minus1

(26)


Δ = Δ1015840= 120575 (27)



Acknowledgments


References


















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









Documents

Research Article Sharp Upper Bounds for the Laplacian ...downloads.hindawi.com/journals/mpe/2013/720854.pdf · the Laplacian spectral radius of irregulargraphs as follows. Let be