On Energy and Laplacian Energy of Graphs

Transcription

1 Electroic Joural of Liear Algebra Volume 31 Volume 31: 016 Article O Eergy ad Laplacia Eergy of Graphs Kikar Ch Das Sugkyukwa Uiversity, kikardas003@googl com Seyed Ahmad Mojalal Sugkyukwa Uiversity, ahmad_mojalal@yahoocom Follow this ad additioal works at: Recommeded Citatio Das, Kikar Ch ad Mojalal, Seyed Ahmad 016, "O Eergy ad Laplacia Eergy of Graphs", Electroic Joural of Liear Algebra, Volume 31, pp DOI: This Article is brought to you for free ad ope access by Wyomig Scholars Repository It has bee accepted for iclusio i Electroic Joural of Liear Algebra by a authorized editor of Wyomig Scholars Repository For more iformatio, please cotact scholcom@uwyoedu

2 ON ENERGY AND LAPLACIAN ENERGY OF GRAPHS KINKAR CH DAS AND SEYED AHMAD MOJALLAL Abstract Let G = V,E be a simple graph of order with m edges The eergy of a graph G, deoted by EG, is defied as the sum of the absolute values of all eigevalues of G The Laplacia eergy of the graph G is defied as LE = LEG = µ i m, where µ 1, µ,, µ, µ = 0 are the Laplacia eigevalues of graph G I this paper, some lower ad upper bouds for EG are preseted i terms of umber of vertices, umber of edges, maximum degree ad the first Zagreb idex, etc Moreover, a relatio betwee eergy ad Laplacia eergy of graphs is give Key words Graph, Spectral radius, Eergy, Laplacia eergy, First Zagreb idex, Determiat AMS subject classificatios 05C50 1 Itroductio The eergyeg ofagraphg, defied asthe sum ofthe absolute values of its eigevalues, belogs to the most popular graph ivariats i chemical graph theory It origiates from the π-electro eergy i the Hückel molecular orbital model, but has also gaied purely mathematical iterest Gutma itroduced this defiitio of the eergy of a simple graph i his paper The eergy of a graph [1] He otes that at first, very few mathematicias seemed to be attracted to the defiitio I the past decade, iterest i graph eergy has icreased ad may differet versios have bee itroduced I 006, Gutma ad Zhou defied the Laplacia eergy of a graph as the sum of the absolute deviatios ie, distace from the mea of the eigevalues of its Laplacia matrix [15] Similar variats of graph eergy were developed for the sigless Laplacia matrix, the distace matrix, the icidece matrix, ad eve for a geeral matrix ot associated with a graph [3] I 010, Cavers, Fallat, ad Kirklad first studied the Normalized Laplacia eergy of a graph, also kow as the Radić eergy for its coectio to the Radić idex [3] Let G = V,E be a simple graph with vertex set VG = {v 1,v,,v } ad Received by the editors o August 10, 014 Accepted for publicatio o March, 016 Hadlig Editor: Brya L Shader Departmet of Mathematics, Sugkyukwa Uiversity, Suwo , Republic of Korea kikardas003@googl com, ahmad mojalal@yahoocom The first author was supported by the Natioal Research Foudatio fuded by the Korea govermet with the grat o 013R1A1A

3 168 KC Das ad SA Mojallal edge set EG, EG = m Let d i be the degree of the vertex v i for i = 1,,, The maximum ad miimum vertex degrees are deoted by ad δ, respectively Let N i be the eighbor set of the vertex v i VG Deote by, ω, the clique umber of graph G If vertices v i ad v j are adjacet, we deote that by v i v j EG The adjacecy matrix AG of G is defied by its etries a ij = 1 if v i v j EG ad 0 otherwise Let λ 1 λ λ λ deote the eigevalues of AG The largest eigevalue λ 1 is called the spectral radius of graph G Whe more tha oe graph is uder cosideratio, we write λ i G istead of λ i Some well kow results are the followig: 11 1 λ i = 0, λ i = m ad deta = λ i Moreover,the spectralradius λ 1 is at least the averagevertexdegree i the graph, that is, 13 λ 1 m with equality holdig if ad oly if G is isomorphic to a regular graph The eergy of the graph G is defied as 14 EG = λ i, where λ i, i = 1,,,, are the eigevalues of graph G For its basic properties, applicatiosicludig variouslowerad upperbouds, see[7,9, 14,17, 19, 0, 1, 9] Maximum ad miimum values of the eergy are kow for various classes of graphs, ad i some cases also the secod-largest/secod-smallest ad further values as well as the correspodig extremal graphs; see the book [0] for recet results ad the refereces cited therei The Laplacia matrix of G is LG = DG AG The Laplacia matrix has oegative eigevalues µ 1 µ µ = 0 Deote by SpecG = {µ 1,µ,,µ } the spectrum of LG, ie, the Laplacia spectrum of G Whe more tha oe graph is uder cosideratio, we write µ i G istead of µ i

4 As well kow [], O Eergy ad Laplacia Eergy of Graphs µ i = m The Laplacia eergy of the graph G is defied as [15] LE = LEG = µ i m 16 For its basic properties, icludig various lower ad upper bouds, see [5, 6, 8, 10, 4, 8] As usual, K ad K 1,, deote, respectively, the complete graph ad the star o vertices For other udefied otatios ad termiology from graph theory, the readers are referred to [1] The paper is orgaized as follows I Sectio, we list some previously kow results I Sectio 3, we preset a lowerboud o eergy EG of graph G I Sectio 4, we obtai a upper boud o eergy EG of graph G I Sectio 5, we give a relatio betwee eergy ad Laplacia eergy of graphs Prelimiaries I this sectio, we list some previously kow results that will be eeded i the ext three sectios Lemma 1 [5] Let B be a p p symmetric matrix ad let B k be its leadig k k submatrix; that is, B k is matrix obtaied from B by deletig its last p k rows ad colums The for i = 1,,, k 1 ρ p i+1 B ρ k i+1 B k ρ k i+1 B, where ρ i B is the i-th largest eigevalue of B Lemma [11] Let B ad C be two real symmetric matrices of size The for ay 1 k, λ i B +C λ i B+ where λ i M is the i-th largest eigevalue of M λ i C, The followig upper boud o λ 1 has bee give i [18]: Lemma 3 [18] Let G be a coected graph of order with m edges The λ 1 m +1 with equality holdig if ad oly if G = K or G = K 1,

5 170 KC Das ad SA Mojallal A d-regulargraph G o vertices is strogly d-regulardeote by srg, d, λ, µ if there exist positive itegers d, λ ad µ such that every vertex has d eighbors, every adjacet pair of vertices has λ commo eighbors, ad every oadjacet pair has µ commo eighbors The followig result is well kow []: are Lemma 4 Let G be isomorphic to srg, d, λ, µ The the eigevalues of G i d of multiplicity 1, ii λ µ+ λ µ +4d µ iii λ µ λ µ +4d µ of multiplicity 1 of multiplicity 1 Corollary 5 Let G be isomorphic to srg, d, λ, µ The Esrg, d, λ, µ = d+ d µ dλ µ λ µ +4d µ [ ] d+λ µ, λ µ +4d µ [ ] + d+λ µ λ µ +4d µ Proof By 14, [ EG = d+ 1 [ + 1 d+λ µ λ µ +4d µ ] + d+λ µ λ µ +4d µ = d+ λ µ +4d µ = d+ d µ dλ µ λ µ +4d µ ] λ µ+ λ µ +4d µ λ µ +4d µ λ µ λ µ[d+λ µ] λ µ +4d µ Corollary 6 Let G be isomorphic to srg, d, λ, λ The Esrg, d, λ, λ = d+ d λ Lemma 7 [4] Let G be a graph with vertex set VG = {v 1, v,, v } The 3 N i N j = d j 1, v i VG, v j V G,j i v j:v iv j EG

6 O Eergy ad Laplacia Eergy of Graphs 171 where d i is the degree of the vertex v i ad N i N j is the cardiality of the commo eighbors of v i ad v j 3 Lower boud for the eergy of graphs I this sectio, we give a lower boud o eergy EG i terms of, m ad the determiat of the adjacecy matrix of graph G First we metio three popular lower bouds o eergy EG of graphs Li et al [0] gave the followig lower boud i terms of m: 31 EG m with equality holdig if ad oly if G cosists of a complete bipartite graph K a,b such that a b = m ad arbitrarily may isolated vertices McClellad [1] obtaied the followig lower boud i terms of, m ad the determiat of the adjacecy matrix of graph G: 3 EG m+ deta / Recetly, Das et al [9] preseted a lower boud for osigular graph ie, a graph for which 0 is ot a eigevalue, ad hece deta > 0 ad the boud is as follows: EG m deta 33 ++l m We ow give a lower boud o EG of graph G i terms of, m ad deta Theorem 31 Let G be a coected graph of order ad m edges The { m EG mi + m 4m +Z, m +1+ } 34 +Z, where 1 det A Z = m +1 ad det A is the determiat of the adjacecy matrix of graph G Moreover, equality holds i 34 if ad oly if G = K 35 Proof From 1 ad 14, we get E G = λ i + 1 i<j = m+λ 1 λ i + i= λ i λ j i<j = m+λ 1 EG λ 1 + λ i λ j i<j λ i λ j

7 17 KC Das ad SA Mojallal By the Arithmetic-Geometric mea iequality, we get i<j λ i λ j = λ i i= / / deta λ 1 that is, Usig iequality 37 i 35, we get det A E 1/ G m+λ 1 EG λ 1 +, E G λ 1 EG [ λ 1 m λ 1 + det A By solvig the above iequality with EG > λ 1, we get EG λ 1 + Usig, we have 38 EG λ 1 + λ 1 m λ 1 + det A λ 1 1 ] det A m λ 1 + m +1 The Let 1 det A fx = x+ m x + m +1 f x x = 1 1 det A m x + m +1 Thus, fx is a icreasig fuctio o x m+ 1 det A m +1 1

8 ad decreasig fuctio o x From 13 ad Lemma 3, we get O Eergy ad Laplacia Eergy of Graphs 173 m+ 1 det A m +1 1 From 38, we get { EG mi f m λ 1 m +1 m, f m +1 }, which gives the required result i 34 This completes the first part of the proof Suppose that equality holds i 34 The all the iequalities i the above must be equalities I particular, from equality i 36, we get λ = λ 3 = = λ From equality i 38, we get G = K or G = K 1,, by Lemma 3 The above result holds for complete graph K Moreover, all other iequalities i the above must be equalities for complete graph K Coversely, let G = K Sice the adjacecy spectrum of K is, 1, 1,, 1, }{{} we have Now, we have m + 1 det A Z = = m +1 m 4m +M = ad m +1+ +M = Hece, the equality holds i 34 for complete graph K Corollary 3 Let G be a coected graph of order ad m edges The EG m + 1 ] deta 39 [1+, m +1

9 174 KC Das ad SA Mojallal where det A is the determiat of the adjacecy matrix of graph G Moreover, equality holds i 39 if ad oly if G = K Proof Agai from 13 ad Lemma 3, we get m λ 1 m +1 with left right equality holdig if ad oly if G is a regular graph G = K or G = K 1, Bearig this i mid, from Theorem 31, we get EG λ 1 + m λ 1 + deta λ 1 1 m 1 deta + +, m +1 which gives the required result i 39 Moreover, equality holds i 39 if ad oly if G = K Corollary 33 Let G be a coected graph of order with m edges ad maximum degree The EG m + [1+ ] deta 310, where det A is the determiat of the adjacecy matrix of graph G Moreover, equality holds i 310 if ad oly if G = K Proof It is well kow that for coected graph G, λ 1 with equality holdig if ad oly if G is a regular graph Usig this result, from the proof of Corollary 3, we get the lower boud i 310 Moreover, equality holds i 310 if ad oly if G = K H 1 H H 3 H 4 Figure 1 Graphs H 1, H, H 3 ad H 4 Example 34 Four graphs H 1, H, H 3 ad H 4 have bee show i Figure 1 The umerical results related to EG ad the bouds that metioed above are

10 O Eergy ad Laplacia Eergy of Graphs 175 listed i the followig These show that our boud 34 is better tha the other three bouds 31, 3 ad 33 for the graphs H 1, H, H 3 ad H 4 We should ote that these results are presetig as rouded to three decimal places G EG H H H H Table 1 The eergy ad the values of the lower bouds 31, 3, 33, ad 34 for the graphs depicted i Figure 1 4 Upper boud for the eergy of graphs I this sectio, we give a upper boud o eergy EG The rak of a matrix B is the maximal umber of liearly idepedet rows or colums of B Let r be the rak of adjacecy matrix AG of graph G The first Zagreb idex M 1 = M 1 G is equal to the sum of squares of the vertex degrees of the graph G [13] First we metio four upper bouds o eergy EG of graphs Koole et al [19] gave the followig upper boud i terms of ad m: EG m + [m ] m 41 BoZhou [9] obtaied the followigupper boud i terms of, m ad the first Zagreb idex M 1 G: M1 G M1 G 4 EG + m Das et al [7] gave the two followig upper bouds: 43 EG m δ+4 m 3 1 1/ω ad 44 EG m δ + [ ] m 4m δ

11 176 KC Das ad SA Mojallal We are ow ready to give a upper boud o eergy EG of graphs G i terms of, m, r, ad M 1 G Theorem 41 Let G be a coected graph of order with m edges, maximum degree ad the first Zagreb idex M 1 G The 45 m m EG + +P, where P = [ ] r 1 8m m 4 M 1 G M 1G m + 4 ad r is the rak of the adjacecy matrix of graph G Moreover, equality holds i 45 if ad oly if G = K or G = K /,/ or G = srg, d, dd 1, dd 1 Proof Sice r is the rak of the adjacecy matrix of graph G, there are exactly r o-zero eigevalues ad hece r We ca assume that λ 1 λ λ r > 0 we have Let us cosider the matrix A G Sice the i, j etry of A G is { di if i = j, N i N j otherwise, 46 λ 4 i = tra4 = d i + N i N j 1 i<j 47 By the Cauchy-Schwarz iequality, we have 1 i<j N i N j 1 i<j N i N j with equality holdig if ad oly if N i N j = N k N l for ay v i, v j v k, v l, i j, k l From 3, we get v i VG v j V G,j i N i N j = v i VG v j:v iv j EG d j 1

12 O Eergy ad Laplacia Eergy of Graphs 177 We deote the average degree of the adjacet vertices of vertex v i of graph G by m i From the above, we get N i N j = d i m i d i Sice 1 i<j v i VG v i VG M 1 G = d i = d i m i ad v i VG d i = m, we get 48 1 i<j N i N j = 1 M 1G m From 46, 47 ad 48, we get 49 r λ 4 i = λ 4 i M 1 1G+ M 1G m with equality holdig i 49 if ad oly if N i N j = N k N l for ay v i, v j v k, v l, i j, k l Sice 410 r λ i = m, ad by the Cauchy-Schwarz iequality, we have r r EG = λ i = λ 1 + λ i + λ i λ j 411 = λ 1 + i= i<j r m λ 1 + i<j r λ i λ j λ 1 + m λ 1 r 1r + i<j r λ i λ j r = λ 1 + m λ 1 + r 1r λi i= r i= λ 4 i

13 178 KC Das ad SA Mojallal Sice with 49 ad 410, we get m λ 1 r [ EG λ 1 + m λ 1 + 1r m λ 1 M 1 G M 1G m +λ 4 1 ] + m m + [ r 1 8m m 4 M 1 G M 1G m + ] 4 41 This completes the first part of the proof Suppose that equality holds i 45 The all the iequalities i the above must be equalities From equality i 41, we get λ 1 = = m Therefore, G is a regular graph Suppose G is d-regular graph Sice r is the rak of the adjacecy matrix of coected graph G, we have r We cosider two cases: i r =, ii r > Case i : r = I this case, G has two o-zero eigevalues Sice λ i = 0, we must have λ 1 = λ, λ = λ 3 = = λ = 0 By 1, we get λ 1 = m = 4m, that is, m = 4 ad λ 1 = / = λ Hece, G = K /,/ Case ii : r > From equality i 411, we get 413 λ = λ 3 = = λ r From equality i 49, we get 414 N i N j = N k N l for ay v i, v j v k, v l, i j, k l

14 O Eergy ad Laplacia Eergy of Graphs 179 If ay v i, v j EG i j, the G = K ad also the equality holds i 413 Otherwise, there exists at least oe v i, v j / EG i j Usig 414, we get λ = N i N j = N k N l = µ for v i, v j EG, v k, v l / EG By Lemma 7 with the above result, we get λ = µ = dd 1 Sice G is d-regular graph, from the above we must have G = srg, d, dd 1, dd 1 Moreover, by Lemma 4, 413 holds for srg, d, dd 1, dd 1 Coversely, we have to prove that the equality holds i 45 for K, K /,/ ad srg, d, dd 1, dd 1 First cosider G = K =, r =, m = ad M 1 K = Thus, we have P = [ ] r 1 8m m 4 M 1 G M 1G m + 4 = =, [ + 4 ] ad hece, + m m +P = = EK Next cosider G = K /,/ We have r =, = / ad m = / The P = 0, ad hece, m m + +P = = EK /,/ Fially, cosider the case that G = srg, d, dd 1 r = ad M 1 G = d By Corollary 5, we get EG = d+ d d, dd 1 Therefore,

15 180 KC Das ad SA Mojallal Now, P = [ ] r 1 8m m 4 M 1 G M 1G m + 4 [ ] = d d d d d 1 +d 4 ad hece, 415 = d [ d d 1 +d ] = d d, m m + +P = d+ d d = EG Corollary 4 Let G be a d-regular coected graph of order The EG d+ d d with equality holdig if ad oly if G = K or G = srg, d, dd 1, dd 1 Proof Sice r ad d i = d i 45, we get the required result i 415 Moreover, equality holds i 415 if ad oly if G = srg, d, dd 1, dd 1 or G = K, by Theorem 41 Corollary 43 Let G be a coected graph of order with m edges, maximum degree ad the first Zagreb idex M 1 G The 416 m m EG + +P, where P = [ 8m m 4 M 1 G M 1G m + ] 4 Moreover, equality holds i 416 if ad oly if G = srg, d, dd 1, dd 1 or G = K Proof Sice r, the result follows from Theorem 41

16 O Eergy ad Laplacia Eergy of Graphs 181 H 5 H 6 H 7 H 8 Figure Graphs H 5, H 6, H 7 ad H 8 G EG H H H H Table The eergy ad the values of the upper bouds 41, 4, 43, 44, ad 45 for the graphs depicted i Fig Example 44 Four graphs H 5, H 6, H 7 ad H 8 have bee show i Figure The umerical results related to EG ad the bouds that metioed above are listed i Table these results are presetig as rouded to three decimal places From the Table, we show that our boud 45 is better tha the other four bouds 41, 4, 43, ad 44 for the graphs H 5, H 6, H 7 ad H 8 5 Relatio betwee eergy ad Laplacia eergy of graphs I this sectio, we give a relatio betwee eergy ad Laplacia eergy of graphs Let σ 1 σ be the largest positive iteger such that 51 µ σ m The from [6], we have LEG = µ i m 5 = S σg 4mσ { = max S i G 4mi } 53, 1 i

17 18 KC Das ad SA Mojallal where σ S σ G = µ i H 9 Figure 3 Graph H 9 For G = K 1, >, For G = H 9, see, [7] LEG = 4+ 4 > = EG LEG < EG From the above, it is easy to see that eergy ad Laplacia eergy are icomparable So it is iterestig to fid a upper boud o LEG EG Recetly, So et al [6] preseted the followig relatio betwee eergy ad Laplacia eergy of graphs with usig Ky Fa Theorem, LEG EG+ d i m 54 We ow give aother relatio betwee eergy ad Laplacia eergy of graphs Theorem 51 Let G be a graph of order with m edges ad vertex degrees d 1, d,, d The σ LEG EG+ d i m 55, where σ is the largest positive iteger satisfyig 51

18 Proof For ay k 1 k, O Eergy ad Laplacia Eergy of Graphs 183 λ i AG = λ i+1, where λ i AG is the i-th largest eigevalue of AG Usig this result with Lemma, we get µ i d i λ i+1 From the defiitio of graph eergy, we have EG = λ i = λ i = { i = max λ i 0 λ i<0λ From the above two results, for ay k, we get µ i } λ i+1 : 1 k λ i+1 d i +EG for ay k, 1 k Sice σ is the largest positive iteger satisfyig 51, from the above, we ca write σ µ i 4mσ σ d i +EG 4mσ Usig 5, from the above, we get the required result i 55 Remark 5 Our result i 55 is always better tha the result i 54 Proof Let ν be the largest positive iteger such that d ν m assume that d 1 d d Thus, we have ν d i m ν = d i mν = m d i mν i=ν+1 = m ν m d i = d i Moreover, d i m ν = d i m i=ν+1 + i=ν+1 i=ν+1 m d i

19 184 KC Das ad SA Mojallal From the above two relatios, we get 56 d i m ν = d i m Now we have to show that ν d i m { = max d i m : 1 k }, that is, 57 d i m ν d i m for ay k, k = 1,,, If k = ν, the the equality holds i 57 Otherwise, k ν We ow cosider two followig cases: Case i : k < ν Sice ν i=k+1 d i m 0 ad ν d i m = d i m + ν i=k+1 d i m, we get the result i 57 we have Case ii : k > ν Agai, sice d i m = ν i=ν+1 d i m d i m + i=ν+1 < 0, d i m < ν d i m which gives the result i 57 Thus, d i m ν = d i m { = max σ d i m d i m : 1 k }

20 58 O Eergy ad Laplacia Eergy of Graphs 185 So et al [6] obtaied a lower boud for bipartite graph as follows: LEG d i m I [16], it has bee provedthe followigresult: For coected graphgwith vertex degrees d 1 d d > 0, we have 59 µ i d i +1 for ay k, 1 k Now we preset a lower boud o LE of graph G that always is better tha the lower boud i 58 Moreover, our result is true for all coected graphs Theorem 53 Let G be a coected graph of order with m edges The LEG + d i m 510 Proof Let ν be the largest positive iteger such that d ν m 59, from 53, we get LEG S ν 4mν ν d i +1 4mν ν = d i m + = + d i m by 56 Usig the result Ackowledgmet The authors are grateful to the aoymous referee for givig several valuable commets ad suggestios REFERENCES [1] JA Body ad USR Murty Graph Theory with Applicatios America Elsevier Publishig Co, Ic, New York, 1976 [] D Cvetković, P Rowliso, ad S Simić A Itroductio to the Theory of Graph Spectra Cambridge Uiversity Press, New York, 010

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