Miskolc Mathematical Notes HU e-issn Bounds for Laplacian-type graph energies. Ivan Gutman, Emina Milovanovic, and Igor Milovanovic

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1 Miskolc Mathematical Notes HU e-issn Vol. 6 (05), No, pp DOI: 0.854/MMN Bouds for Laplacia-type graph eergies Iva Gutma, Emia Milovaovic, ad Igor Milovaovic

2 Miskolc Mathematical Notes HU e-issn Vol. 6 (05), No., pp BOUNDS FOR LAPLACIAN TYPE GRAPH ENERGIES IVAN GUTMAN, EMINA MILOVANOVIĆ, AND IGOR MILOVANOVIĆ Received 4 February, 04 Abstract. Let G be a udirected simple ad coected graph with vertices. 3/ ad m edges. Deote by > D 0,, ad > D 0, respectively, the Laplacia, sigless Laplacia, ad ormalized Laplacia eigevalues of G. The Laplacia eergy, sigless Laplacia eergy, ad ormalized Laplacia eergy of G are defied as LE D P m P i, SLE D m i, ad NLE D P j i j, respectively. Lower bouds for LE, SLE, ad NLE are obtaied. 00 Mathematics Subject Classificatio: 05C50; 05C07 Keywords: eergy (of graph), Laplacia eergy, sigless Laplacia eergy, ormalized Laplacia eergy, Radić eergy. INTRODUCTION Let G be a udirected simple ad coected graph with vertices. / ad m edges, ad let d ;d ;:::;d be its vertex degrees. If the i-th ad j -th vertex of the graph G are adjacet, we write i j. The the adjacecy matrix A D.a ij / of G is defied as 8 < if i 6D j ad i j a ij D : 0 otherwise: The eigevalues of A form the (ordiary) spectrum of G; for details o the respective spectral theory see [9]. Deote by D the diagoal matrix of the vertex degrees of G. The Laplacia matrix of G is L D D A ad its eigevalues are > D 0 (see [3, 6, 5]). I additio, Q D D C A is the sigless Laplacia matrix of G ad its eigevalues will be deoted by 0 [0, ]. Because the graph G is assumed to be coected, it has o isolated vertices (i.e., d i > 0 for all i ) ad therefore the matrix D = is well defied. The L D D = LD = is called the ormalized Laplacia matrix of the graph G. Its c 05 Miskolc Uiversity Press

3 96 IVAN GUTMAN, EMINA MILOVANOVIĆ, AND IGOR MILOVANOVIĆ eigevalues are > D 0. For details of the spectral theory of the ormalized Laplacia matrix see [8]. It is coveiet to write the ormalized Laplacia matrix as I R, where R is the so-called Radić matrix [4, 9, 30], whose.i;j /-etry is ( p = di d j if i 6D j ad i j r ij D 0 otherwise: The (ordiary) eergy of the graph G is defied as [3] E D E.G/ D j i j : (.) Its theory is owadays well elaborated [3]. Eergy like spectral ivariats have bee itroduced also for other graph matrices [8]. I this paper we are cocered with the Laplacia [, 3], sigless Laplacia [], ad ormalized Laplacia (or Radić) eergies [5, 0], defied as LE D LE.G/ D m i SLE D SLE.G/ D NLE D NLE.G/ D i j i respectively. I what follows lower bouds for LE, SLE ad NLE are obtaied. Remark. I aalogy to (.), the Radić eergy is defied as the sum of the absolute values of the eigevalues of the Radić matrix. It has bee show i [0], that the Radić eergy coicides with the ormalized sigless Laplacia eergy. Remark. Oe could also cosider the ormalized sigless Laplacia matrix, D = QD = ad its eergy (sum of absolute values of eigevalues). However, the eergy of this matrix is exactly the same as the ormalized Laplacia eergy, NLE [0]. For the geeral defiitio of the eergy of a matrix see [8]. The Laplacia, sigless Laplacia, ad ormalized (or Radić) Laplacia spreads of a graph G are defied as LS.G/ D, SLS.G/ D, ad NLS.G/ D, respectively (see [5, 3, 5, 4]).. PRELIMINARIES I this sectio we recall some results from spectral graph theory, ad state a few aalytical iequalities eeded for our work. m j

4 LAPLACIAN TYPE GRAPH ENERGIES 97 Lemma ([3]). Let G be a udirected simple ad coected graph with ;, vertices ad m edges. The i D d i D m ad i D di C d i D M C m where M is the sum of squares of the vertex degrees, usually referred to as the first Zagreb idex (see [, 7, 9]). Lemma ([]). Let G be a udirected simple ad coected graph with ;, vertices ad m edges. The r M m M 4m : (.) Lemma 3 ([3]). Let G a.;m/-graph, such that 3 ad m. The LE.G/ C m (.) with equality if ad oly if D 3 or for 4 if D D D m. Lemma 4 ([6]). Let G be a udirected simple ad coected graph with ; 3, vertices ad m edges. The r q LS.G/ D./.M C m/ 4m : (.3) Equality holds if ad oly if G Š K. Lemma 5 ([7]). Let a ;a ;:::;a be real umbers ad p ;p ;:::;p o-egative real umbers with the property p C p C C p D. The, for each, 0 ad,! p i aį p i a i : (.4) For the case 0, the opposite iequality is valid. Equality i (.4) holds if ad oly if D 0 or D or a D a D D a. Lemma 6 ([6]). Let a ;a ;:::;a be real umbers, ad assume that there are r;r R such that < r a i R < C, for each i D ;;:::;. The for ay o-egative p ;p ;:::;p with the property p C p C C p D, 0 p i ai The costat is sharp.! p i a i.r r/ p i a i p i a i: (.5)

5 98 IVAN GUTMAN, EMINA MILOVANOVIĆ, AND IGOR MILOVANOVIĆ Lemma 7 ([3]). Let G be a udirected simple ad coected graph with ;, vertices ad m edges. The where R D P ij i D ad i D C R (.6) d i d j ; for details o the graph ivariat R see [4, ]. Lemma 8 ([7]). Let G be a udirected simple ad coected graph with ;, vertices ad m edges. The i D d i D m ad D di C d i D M C m where M is the first Zagreb idex. i Lemma 9 ([7]). The sigless Laplacia spread has a upper boud s Œ.M C m/ 4m SLS.G/ : Lemma 0 ([4]). Suppose that G is a graph without isolated vertices. The q./.m C M / 4m : (.7) 3. MAIN RESULTS 3.. Lower boud for Laplacia eergy Theorem. Let G be a udirected coected graph with, 3, vertices ad m edges. The LE.G/ m C q./.m C M / 4m : (3.) Proof. Iequality (3.) directly follows from iequalities (.) ad (.7). Corollary. Let G be a udirected graph with, 3, vertices ad m edges. The LE.G/ m C r m../ m/ : Corollary. Let G be a udirected simple ad coected k-regular graph with, 3, vertices ad m edges, < k. The LE.G/ k C p k. k / :

6 LAPLACIAN TYPE GRAPH ENERGIES 99 Theorem. Let G be a udirected simple ad coected graph with ; 3 vertices ad m edges. The r q LE.G/./.M C m/ 4m : (3.) Proof. For ad p i WD, a i WD i, i D ;;:::, r WD ad R WD, the iequality (.5) trasforms ito X i./! i./ i i.e., based o Lemma,./.M C m/ 4m. X / i Sice i m i m D LE.G/ usig iequality (.3), from the above iequality we obtai (3.). X i m : Usig Lemma, we arrive at the followig.;m/-type lower boud for the Laplacia eergy: Corollary 3. Let G be a udirected simple ad coected graph with ; 3, vertices ad m edges. The s 4m../ m/ LE.G/ : (3.3)./ Corollary 4. Let G be a udirected simple ad coected k-regular graph with ; 3, vertices ad m edges, < k. The r k. k / LE.G/ > : Remark 3. Sice for udirected k-regular graphs, LE D E, the iequality i Corollary 4 provides a lower boud also for the ordiary eergy. Iequalities (3.) ad (3.) are icomparable. Thus, for example, if G Š K, the iequality (3.) is stroger tha (3.), but if G Š K ;, 8, the the opposite is valid.

7 00 IVAN GUTMAN, EMINA MILOVANOVIĆ, AND IGOR MILOVANOVIĆ 3.. Lower boud for sigless Laplacia eergy Theorem 3. Let G be a udirected simple ad coected graph with, 3, vertices ad m edges. The s..m C m/ 4m SLE.G/ / : (3.4) Proof. For p i WD, a i D i, i D ;;:::;, r D ad R D, the iequality (.5) becomes! i i m i : Bearig i mid Lemma 8, the above iequality becomes.m C m/ 4m SLS.G/ SLE.G/ : By Lemma 9 ad the above iequality, we obtai (3.4). Bearig i mid Lemma ad iequality (3.4), we arrive at a lower boud for SLE.G/ depedig oly o the parameter m. Corollary 5. Let G be a udirected simple ad coected graph with ; 3, vertices ad m edges. The SLE.G/ p m : Corollary 6. Let G be a udirected simple ad coected graph with ; 3, vertices ad m edges, which is k-regular, < k. The SLE.G/ p k : 3.3. Lower boud for ormalized Laplacia eergy Theorem 4. Let G be a udirected simple ad coected graph with ; 3, vertices ad m edges. Let, as before, R D P. The d i d j r p NLS.G/. /R : (3.5) Equality holds if ad oly if G Š K. Proof. Accordig to (.6) we have that ij./. C R / D./ D X i<j i! i. i j / : (3.6)

8 LAPLACIAN TYPE GRAPH ENERGIES 0 By Lemma 5, i.e., by iequality (.4), for D ad D, we get for each i D ;3;:::; X i<j. i / C. i /. / (3.7). The,. i j / Œ. i / C. i / C. / id 3. / C. / D. / which combied with (3.6) yields./. C R / D./R. / from which the iequality (3.5) follows. Equality i (3.7) holds if ad oly if D D D. Therefore, equality i (3.5) holds if ad oly if G Š K. This completes the proof of Theorem 4. Corollary 7. Let G be a udirected simple ad coected k-regular graph, < k, with ; 3, vertices ad m edges. The s. k / NLS.G/ :./k Equality holds if ad oly if k D, i.e., G Š K. We ow state a theorem, aalogous to Theorem, which provides a lower boud for NLE i terms of parameters ad R. Theorem 5. Let G be a udirected simple ad coected graph with ; 3, vertices ad m edges. The r p NLE.G/./R : (3.8) Proof. For WD, p i WD, a i WD i, i D ;;:::;, r D ad R D, iequality (.5) becomes! X i X./ i X X./ i i: Havig i mid Lemma 7, the above iequality trasforms ito./. C R / NLS.G/ i : (3.9)

9 0 IVAN GUTMAN, EMINA MILOVANOVIĆ, AND IGOR MILOVANOVIĆ Sice accordig to (3.9) we obtai i j i./. C R / NLS.G/NLE.G/ : (3.0) Combiig (3.5) ad (3.0) we arrive at (3.8). Remark 4. For a k-regular graph, R D m=k D =.k/. Sice for k-regular graphs, NLE D k E D LE, iequality (3.8) is equivalet to the result prove i k Corollary 4. REFERENCES [] N. Abreu, D. M. Cardoso, I. Gutma, E. A. Martis, ad R. M., Bouds for the sigless Laplacia eergy, Liear Algebra Appl., vol. 435, pp , 0. [] V. Adova, S. Bogoev, D. Dimitrov, M. Pilipczuk, ad R. Škrekovski, O the Zagreb idex iequality of graphs with prescribed vertex degrees, Discr. Appl. Math., vol. 59, pp , 0. [3] N. L. Biggs, Algebraic Graph Theory. Cambridge: Cambridge Uiv. Press., 974. [4] S. B. Bozkurt, A. D. Gügör, I. Gutma, ad A. S. Çevik, Radić matrix ad Radić eergy, MATCH Commu. Math. Comput. Chem., vol. 64, pp , 00. [5] M. Cavers, S. Fallat, ad S. Kirklad, O the ormalized Laplacia ad geeral Radić idex of graphs, Liear Algebra Appl., vol. 33, pp. 7 90, 00. [6] P. Ceroe ad S. S. Dragomir, A refiemet of the Grüss iequality ad applicatios, Tamkag J. Math., vol. 38, o., pp , 007. [7] S. Che ad W. Liu, Extremal Zagreb idices of graphs with a give umber of cut edges, Graphs Combi., vol. 30, pp. 09 8, 04. [8] F. R. K. Chug, Spectral Graph Theory. Providece: Am. Math. Soc., 997. [9] D. Cvetković, M. Doob, ad H. Sachs, Spectra of Graphs Theory ad Applicatio. New York: Academic Press, 980. [0] D. Cvetković, P. Rowliso, ad S. Simi c, Sigless Laplacia of fiite graphs, Liear Algebra Appl., vol. 43, pp. 55 7, 007. [] D. Cvetković ad S. Simić, Towards a spectral theory of graph based o the sigless Laplacia ii, Liear Algebra Appl., vol. 43, pp , 00. [] C. S. Edwards, The largest vertex degree sum for a triagle i a graph, Bull. Lodo Math. Soc., vol. 9, pp , 977. [3] Y. Fa, J. Xu, Y. Wag, ad D. Liag, The Laplacia spread of a tree, Discr. Math. Theoret. Comput. Sci., vol. 0, pp , 008. [4] G. H. Fath-Tabar ad A. R. Ashrafi, Some remarks o Laplacia eigevalues ad Laplacia eergy of graphs, Math. Commu., vol. 5, o., pp , 00. [5] H. Gomes, I. Gutma, E. A. Martis, M. Robbiao, ad B. Sa Martí, O Radić spread, MATCH Commu. Math. Comput. Chem., vol. 7, pp , 04. [6] R. Groe ad R. Merris, The Laplacia spectrum of a graph ii, SIAM J. Discrete Math., vol. 7, pp. 9, 994. [7] A. D. M. Gügör, A. S. Çevik, ad N. Habibi, New bouds for the spread of the sigless Laplacia spectrum, Math. Ieq. Appl., i press. j

10 LAPLACIAN TYPE GRAPH ENERGIES 03 [8] I. Gutma, Comparative studies of graph eergies, Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur., vol. 44, pp. 7, 0. [9] I. Gutma ad K. C. Das, The first Zagreb idex 30 years after, MATCH Commu. Math. Comput. Chem., vol. 50, pp. 83 9, 004. [0] I. Gutma, B. Furtula, ad B. Bozkurt, O Radić eergy, Liear Algebra Appl., vol. 44, pp , 04. [] I. Gutma ad B. Zhou, Laplacia eergy of graph, Liear Algebra Appl., vol. 44, pp. 9 37, 006. [] X. Li ad I. Gutma, Mathematical Aspects of Radić Type Molecular Structure Descriptors. Kragujevac: Uiv. Kragujevac, 006. [3] X. Li, Y. Shi, ad I. Gutma, Graph Eergy. New York: Spriger, 0. [4] M. H. Liu ad B. L. Liu, Sigless Laplacia spread, Liear Algebra Appl., vol. 309, pp , 009. [5] R. Merris, Laplacia matrices of graphs: A survey, Liear Algebra Appl., vol , pp , 994. [6] I. v. Milovaović ad E. I. Milovaović, Remark o iequalities for the Laplacia spread of graphs, Czech. Math. J., i press. [7] D. S. Mitriović, J. E. Pečari c, ad A. M. Fik, Classical ad New Iequalities i Aalysis. Dordrecht: Kluwer, 993. [8] V. Nikiforov, The eergy of graphs ad matrices, J. Math. Aal. Appl., vol. 36, pp , 007. [9] J. A. Rodríguez, A spectral approach to the Radić idex, Liear Algebra Appl., vol. 400, pp , 005. [30] J. A. Rodríguez ad J. M. Sigarreta, O the Radić idex ad coditioal parameters of a graph, MATCH Commu. Math. Comput. Chem., vol. 54, pp , 005. [3] B. Zhou ad I. Gutma, Nordhaus Gaddum type relatios for the eergy ad Laplacia eergy of graphs, Bull. Acad. Serbe Sci. Arts (Cl. Math. Natur.), vol. 34, pp., 007. [3] P. Zumstei, Compariso of spectral methods through the adjacecy matrix ad the Laplacia of a graph, ser. Th. Diploma. Zurich: ETH Zürich, 005. Authors addresses Iva Gutma Uiversity of Kragujevac, Faculty of Sciece, P. O. Box 60, Kragujevac, Serbia address: gutma@kg.ac.rs Emia Milovaović Uiversity of Niš, Faculty of Electroics Egieerig, A. Medvedeva 4, 8000 Niš, Serbia address: ema@elfak.i.ac.rs Igor Milovaović Uiversity of Niš, Faculty of Electroics Egieerig, A. Medvedeva 4, 8000 Niš, Serbia address: igor@elfak.i.ac.rs