Laplacian energy of a graph

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1 Liear Algebra ad its Applicatios 414 (2006) Laplacia eergy of a graph Iva Gutma a,, Bo Zhou b a Faculty of Sciece, Uiversity of Kragujevac, Kragujevac, P.O. Box 60, Serbia ad Moteegro b Departmet of Mathematics, South Chia Normal Uiversity, Guagzhou , PR Chia Received 12 July 2005; accepted 14 September 2005 Available olie 27 October 2005 Submitted by R.A. Brualdi Abstract Let G be a graph with vertices ad m edges. Let λ 1,λ 2,...,λ be the eigevalues of the adjacecy matrix of G, ad let µ 1,µ 2,...,µ be the eigevalues of the Laplacia matrix of G. A earlier much studied quatity E(G) = λ i is the eergy of the graph G. We ow defie ad ivestigate the Laplacia eergy as LE(G) = µ i 2m/. There is a great deal of aalogy betwee the properties of E(G) ad LE(G), but also some sigificat differeces Elsevier Ic. All rights reserved. AMS classificatio: 05C50; 15A18; 05C90 Keywords: Laplacia graph spectrum; Graph spectrum; Eergy (of graph); Laplacia eergy (of graph) 1. Itroductio I this paper we are cocered with simple graphs. Let G be such a graph, possessig vertices ad m edges. I what follows we say that G is a (, m)-graph. Let d i be the degree of the ith vertex of G, i = 1, 2,...,. The spectrum of the graph G, cosistig of the umbers λ 1,λ 2,...,λ, is the spectrum of its adjacecy matrix [3]. The Laplacia spectrum of the graph G, cosistig of the umbers µ 1,µ 2,...,µ, is the spectrum of its Laplacia matrix [5,6,13,14]. Correspodig author. Fax: addresses: gutma@kg.ac.yu (I. Gutma), zhoubo@scu.edu.c (B. Zhou) /$ - see frot matter ( 2005 Elsevier Ic. All rights reserved. doi: /j.laa

2 30 I. Gutma, B. Zhou / Liear Algebra ad its Applicatios 414 (2006) The ordiary ad Laplacia graph eigevalues obey the followig well-kow relatios: λ i = 0; µ i = 2m; λ 2 i = 2m, (1) µ 2 i = 2m + di 2. (2) Furthermore, if the graph G has p compoets (p 1), ad if the Laplacia eigevalues are labelled so that µ 1 µ 2 µ, the µ i = 0 for i = 0,...,p 1 ad µ p > 0. (3) The eergy of the graph G is defied as E(G) = λ i. (4) This quatity has a log kow chemical applicatio; for details see the surveys [7 9]. Recetly much work o graph eergy appeared also i the mathematical literature (see, for istace, [1,15 20]). The followig properties of the eergy of a graph will be eeded (for comparative purposes): Note 1 (a) E(G) 0; equality is attaied if ad oly if m = 0. (b) If the graph G cosists of (discoected) compoets G 1 ad G 2, the E(G) = E(G 1 ) + E(G 2 ). (c) If oe compoet of the graph G is G 1 ad all other compoets are isolated vertices, the E(G) = E(G 1 ). Note 2 [12]. E(G) 2m with equality holdig if ad oly if G is regular of degree 0or1. Note 3 [10,11] E(G) 2m + ( ) ] 2m 2 ( 1) [2m (5) with equality holdig if ad oly if G is either a regular graph of degree 0, 1, or 1, or a o-complete coected strogly regular graph with two o-trivial eigevalues both havig absolute value [2m (2m/) 2 ]/( 1).

3 I. Gutma, B. Zhou / Liear Algebra ad its Applicatios 414 (2006) Note 4 [2]. 2 m E(G) 2m. IfG has o isolated vertices, the the equality E(G) = 2 m holds if ad oly if G is a complete bipartite graph. If G has o isolated vertices, the the equality E(G) = 2m holds if ad oly if G is regular of degree The Laplacia eergy cocept Our itetio is to coceive a graph-eergy-like quatity, that istead of Eq. (4) would be defied i terms of Laplacia eigevalues, ad that hopefully would preserve the mai features of the origial graph eergy. Bearig i mid relatios (1) ad (2), we first itroduce the auxiliary eigevalues γ i,i = 1, 2,...,, defied via γ i = µ i 2m. (6) The, i aalogy with Eq. (1)wehave γ i = 0; γi 2 = 2M, (7) where M = m + 1 ( d i 2m ) 2. (8) 2 Recall that 2m/ is the average vertex degree. Cosequetly, M = m if ad oly if G is regular, ad M>motherwise. Defiitio. If G is a (, m)-graph, ad its Laplacia eigevalues are µ 1,µ 2,...,µ, the the Laplacia eergy of G, deoted by LE(G), is equal to γ i, i.e., LE(G) = µ i 2m. (9) That the above defiitio is well chose is see from the followig bouds, that should be compared with Notes 2 4: LE(G) 2M, (10) LE(G) 2m + ( ) ] 2m 2 ( 1) [2M, (11) 2 M LE(G) 2M. (12) I the subsequet sectio we discuss Eqs. (10) (12) i more detail ad provide proofs thereof.

4 32 I. Gutma, B. Zhou / Liear Algebra ad its Applicatios 414 (2006) The mai results Lemma 1. If the graph G is regular, the LE(G) = E(G). Proof. If a (, m)-graph is regular of degree r, the r = 2m/ ad [3] µ i 2m/ = λ i+1, i = 1, 2,..., (13) ad Lemma 1 follows from (9). Theorem 2. Iequality (10) holds for ay (, m)-graph G, where M is give via Eq. (8). Equality is attaied if ad oly if G is either regular of degree 0 or cosists of α copies of complete graphs of order k ad (k 2)α isolated vertices, α 1,k 2. (Recall that i the case k = 2, G is regular of degree 1.) Proof. Cosider the sum S = ( γ i γ j ) 2. (14) j=1 By direct calculatio ( S = 2 γi 2 2 γ i ) γ j = 4M 2LE(G) 2. j=1 Sice S 0, we have 4M 2LE(G) 2 0, which directly leads to Eq. (10). Cosider ow the graphs for which LE(G) = 2M, i.e., for which S = 0. From (14) it is evidet that S = 0 if ad oly if all γ i -values are mutually equal. By (3) ad (6), γ = 2m/. Cosequetly, γ i { 2m/, +2m/} for all i = 1, 2,...,. The from (6) we coclude that G has at most two distict Laplacia eigevalues: 0 ad 4m/. If all Laplacia eigevalues of G are equal to zero, the G has o edges, i.e., G is regular of degree 0. The M = LE(G) = 0 ad the equality i Eq. (10) holds. Suppose, therefore, that G has exactly two distict Laplacia eigevalues. A coected graph has exactly two distict Laplacia eigevalues if ad oly if its diameter is equal to uity, i.e., if it is a complete graph, cf. Theorem 2.5 i [13]. Therefore, our graph G must cosist of compoets that are mutually isomorphic complete graphs (say, of order k) ad isolated vertices. Let G cosist of α compoets isomorphic to K k ad β isolated vertices. The the Laplacia spectrum of G cosists of umbers k [(k 1)α times] ad 0 [α + β times]. Because = kα + β ad 2m = k(k 1)α,itis { k k(k 1)α/(kα + β) for i = 1, 2,...,(k 1)α, γ i = k(k 1)α/(kα + β) fori = (k 1)α + 1,...,kα+ β.

5 I. Gutma, B. Zhou / Liear Algebra ad its Applicatios 414 (2006) Now, i order that all γ i -values be mutually equal, it must be k(k 1)α k kα + β i.e., β = (k 2)α. = k(k 1)α kα + β Theorem 3. LetGbea(, m)-graph ad p, the umber of its compoets (p 1). The LE(G) 2m p + ( ) ] 2m 2 ( p) [2M p. (15) For p = 1 equality i (15) is attaied for the graphs specified i Note 3, but also for other graphs. For p =, G cosists of isolated vertices, LE(G) = 0, ad equality i (15) holds i a trivial maer. For ay p, equality i (15) holds for the graphs cosistig of α copies of complete graphs of order k ad (k 2)α isolated vertices, α 1,k 2, provided (k 1)α = p. (Recall that i this case p = /2; if k = 2, the G is regular of degree 1.) Proof. If G has p compoets, the accordig to (3) ad (6), γ i = 2m/ for i = 0,...,p 1. Bearig this i mid, cosider the o-egative sum p p S = ( γ i γ j ) 2. (16) j=1 I a aalogous maer as i the proof of Theorem 2, we arrive at 2( p)[2m p(2m/) 2 ] 2[LE(G) p(2m/)] 2 0, from which Eq. (15) follows. With regard to the graphs for which LE(G) = 2m p + ( ) ] 2m 2 ( p) [2M p (17) we first observe that ay regular graph for which (5) is a equality, also satisfies (17). These graphs are specified i Note 3. Usig the same reasoig as i the proof of Theorem 2, we coclude that if (17) holds, the G has at most three distict Laplacia eigevalues. If G has oly two distict Laplacia eigevalues, the i a same maer as i the proof of Theorem 2 we arrive at the coclusio that G cosists of α copies of complete graphs of order k ad (k 2)α isolated vertices, α 1,k 2, provided (k 1)α = p. If the graph satisfyig (17) has three distict Laplacia eigevalues the, for istace, it may cosist of α 1 copies of complete graphs of order k 1, α 2 copies of complete graphs of order k 2, ad β isolated vertices, provided k 1 >k 2 2,α 1,α 2 1, ad β,

6 34 I. Gutma, B. Zhou / Liear Algebra ad its Applicatios 414 (2006) β = (k2 1 2k 1 k 1 k 2 )α 1 + (k 2 2 2k 2 k 1 k 2 )α 2 k 1 + k 2 is a o-egative iteger. (Examples: G = K 6 K 2 ad G = 9K 7 9K 2 7K 1.) The characterizatio of all (m, )-graphs for which (17) holds seems to be difficult ad remais a task for the future. Relatio (11) is the special case of (15) for p = 1. Settig p = 0 ito the righthad side of (15) we obtai (10). It ca be show that the right-had side of (15)isa mootoously decreasig fuctio of the parameter p. I particular, the upper boud (11) is better tha (10). Theorem 4. The left-had side iequality (12) holds for ay (, m)-graph, where M is give via Eq. (8). Equality LE(G) = 2 M is attaied if ad oly if G is the complete bipartite graph K /2,/2. The right-had side iequality (12) holds for graphs without isolated vertices. For such graphs, the equality LE(G) = 2M is attaied if ad oly if G is regular of degree 1. For graphs without edges, LE(G) = M = 0 ad (12) is satisfied i a trivial maer. Therefore i what follows we assume that m>0 ad thus 2. Proof of the left-had side iequality. From ( γ i ) 2 = 0 by takig ito accout (7) ad the fact that M>0, we obtai 2M = 2 γ i γ j = 2 γ i γ j i<j i<j ad thus 2M 2 γ i γ j. (18) i<j Now, by (7) ad (9), LE(G) 2 = 2M + 2 γ i γ j, i<j which combied with (18) yields LE(G) 2 4M. It is easy to check that equality i (18) is obeyed by graphs with two vertices. Therefore we assume that 3. Equality i (18) holds if ad oly if there is at most oe positive-valued ad at most oe egative-valued γ i, i.e.,

7 I. Gutma, B. Zhou / Liear Algebra ad its Applicatios 414 (2006) γ 1 > 0,γ 2 = =γ 1 = 0,γ = 2m < 0. (19) From (19) ad the fact that 3 follows that µ 1 = 2m/. Note that coditios (19) are ot satisfied by complete graphs with more tha two vertices, for which µ 1 = /= 2m/. IfG is ot a complete graph, the by [4], δ µ 1, where δ is the miimum vertex degree of G. O the other had, δ 2m/ implies that G is a regular graph, ad the M = m. The by Lemma 1 ad Note 4, LE(G) = 2 M if ad oly if G is the complete bipartite graph K /2,/2. Proof of the right-had side iequality. We start with relatio (10). For a graph with m edges ad o isolated vertex, 2m. Therefore, LE(G) 2M 2M(2m) = 2 Mm. Because M m, we arrive at Mm M. All iequalities occurrig i the above reasoig become equalities i the case of regular graphs of degree 1. Cosequetly, LE(G) = 2M holds for regular graphs of degree 1. For all other graphs M = m ad = 2m caot hold at the same time. Therefore LE(G) = 2M holds oly for regular graphs of degree 1. By this the proof of Theorem 4 is completed. 4. Dissimilarities betwee eergy ad Laplacia eergy I Note 1 three elemetary properties of the graph eergy are stated. Of these, oly (a) has its direct aalog for Laplacia eergy. Ideed, from (9) is evidet that LE(G) 0 ad we already kow (from the proof of Theorem 2) that LE(G) = 0 if m = 0. If the graph G has at least oe edge, the γ = 2m/ is o-zero ad, cosequetly, LE(G) > 0. Observatio 5 (To be compared with Note 1(b)). If the graph G cosists of (discoected) compoets G 1 ad G 2, ad if G 1 ad G 2 have equal average vertex degrees, the LE(G) = LE(G 1 ) + LE(G 2 ). Otherwise, the latter equality eeds ot be satisfied. Proof. Let G, G 1, ad G 2 be (, m)-, ( 1,m 1 )-, ad ( 2,m 2 )-graphs, respectively. The from 2m 1 / 1 = 2m 2 / 2 follows 2m/ = 2m i / i,i = 1, 2, implyig LE(G) = γ i 2m 1 = γ i 2m γ i 2m 2 =LE(G 1 ) + LE(G 2 ). 1 i= 1 +1 If the coditio 2m 1 / 1 = 2m 2 / 2 is ot obeyed, the it may be either LE(G) > LE(G 1 ) + LE(G 2 ) or LE(G) < LE(G 1 ) + LE(G 2 ) or, exceptioally, LE(G) = 2

8 36 I. Gutma, B. Zhou / Liear Algebra ad its Applicatios 414 (2006) LE(G 1 ) + LE(G 2 ). It remais as a ope problem to characterize the graphs satisfyig each of these relatios. Cosider ow a graph G cosistig of a ( 1,m)-graph G 1 (which, i additio, may have p compoets, p 1) ad of additioal 2 isolated vertices. The LE(G) = 1 p µ i(g 1 ) 2m (p + 2m 2). 2 Observatio 6 (To be compared with Note 1(c)). If 2 is sufficietly large, the LE(G) = 4m p + 2 < 4m. (20) Thus, i this case the Laplacia eergy is idepedet of ay other structural features of the graph G. I additio, lim LE(G) = 4m. 2 Proof. For i = 1,..., 1 p, ad sufficietly large 2, µ i > 2m ad therefore 1 p ( LE(G) = µ i 2m ) 2m + (p + 2 ), which, i view of 1 p µ i = 2m, is trasformed ito Eq. (20). What is sufficietly large i the above observatio remais aother ope problem, to be resolved i the future. Refereces [1] R. Balakrisha, The eergy of a graph, Liear Algebra Appl. 387 (2004) [2] G. Caporossi, D. Cvetković, I. Gutma, P. Hase, Variable eighborhood search for extremal graphs. 2. Fidig graphs with extremal eergy, J. Chem. Iform. Comput. Sci. 39 (1999) [3] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs Theory ad Applicatio, third ed., Joha Ambrosius Barth Verlag, Heidelberg, Leipzig, [4] M. Fiedler, Algebraic coectivity of graphs, Czechoslovak Math. J. 23 (1973) [5] R. Groe, R. Merris, The Laplacia spectrum of a graph II,SIAM J. Discrete Math. 7 (1994) [6] R. Groe, R. Merris, V.S. Suder, The Laplacia spectrum of a graph, SIAM J. Matrix Aal. Appl. 11 (1990) [7] I. Gutma, Total π-electro eergy of bezeoid hydrocarbos,topics Curr. Chem. 162 (1992) [8] I. Gutma, The eergy of a graph: old ad ew results, i: A. Bette, A. Kohert, R. Laue, A. Wasserma (Eds.), Algebraic Combiatorics ad Applicatios, Spriger-Verlag, Berli, 2001, pp

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