Lower Bounds of the Kirchhoff and Degree Kirchhoff Indices

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1 SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 7, (205), Lower Bouds of the Krchhoff ad Degree Krchhoff Idces I. Ž. Mlovaovć, E. I. Mlovaovć, E. Glogć Paper dedcated to memory of Vera Nkolć-Staojevć, Pofessor Emertus at the State Uversty of Nov Pazar Abstract: Let G be a udrected coected graph wth, 3, vertces ad m edges. If µ µ 2 µ > µ = 0 ad ρ ρ 2 ρ > ρ = 0 are the Laplaca ad the ormalzed Laplaca egevalues of G, the the Krchhoff ad the degree Krchhoff dces obey the relatos K f (G) = µ ad DK f (G) = 2m ρ, respectvely. The equaltes that determe lower bouds for some varats of G, that cota K f (G) ad DK f (G), are obtaed ths paper. Lower bouds for K f (G) ad DK f (G), kow the lterature, are obtaed as a specal case. Keywords: Krchhoff dex, Degree Krchhoff dex, Laplaca spectrum (of graph), ormalzed Laplaca spectrum (of graph). Itroducto Let G be udrected coected graph wth, 3, vertces ad m edges, wth vertex degree sequece = d d 2 d = δ > 0. Deote by A the adjacecy matrx of the graph G ad by D the dagoal matrx of ts vertex degrees. The L = D A s Laplaca matrx of G. Deote by µ µ 2 µ > µ = 0 egevalues of L (see [8, 8]). Because the graph G s assumed to be coected t has o solated vertces ad therefore the matrx D /2 s well defed. The L = D /2 LD /2 s ormalzed Laplaca matrx of the graph G. Its egevalues are ρ ρ 2 ρ > ρ = 0. For detals of spectral theory of the ormalzed Laplaca matrx see, for example, [6]. For Laplaca ad ormalzed Laplaca egevalues the followg equaltes are vald µ = d = 2m, µ 2 = d 2 + d = M + 2m () Mauscrpt receved May 2, 204; accepted October 4, 204. I. Ž. Mlovaovć, E. I. Mlovaovć are wth the Faculty of Electrcal Egeerg, Nš, Serba; E. Glogć s wth the State Uversty of Nov Pazar, Nov Pazar, Serba; 25

2 26 I. Ž. Mlovaovć etall. ad ρ =, ρ 2 = + 2R, (2) where M s the frst Zagreb dex (see [4]), ad R s Radć dex (see [4, 24]). Graph varat amed Krchhoff dex, s defed as [3] ad the degree Krchhoff dex as [5] K f (G) = µ DK f (G) = 2m ρ The graph varats K f ad DK f are curretly much studed the mathematcal ad mathematco chemcal lterature; see the recet papers [, 9, 0,, 2, 5, 6, 20, 2] ad the refereces cted there. I a few cases these varats ca be determed a closed form. Therefore, the equaltes that gve upper or lower bouds of these varats are otable. The bouds ca be determed terms of usual structural parameters, such as umber of vertces, umber of edges, vertex degrees, ad smlar, or extremal Laplaca ad ormalzed Laplaca egevalues, or Zagreb or Radć dex, etc. (see [2, 3, 9, 0, 5, 6, 7, 20, 22, 23]. I ths paper we cosder lower bouds for some graph varats that, a specal case, reduce to K f (G) ad DK f (G). 2 Ma result We frst gve two equaltes for o-egatve real umbers. Theorem Let a,a 2,...,a be o-egatve real umbers wth the property a +a a = A > 0. I addto, let l,r > 0 ad p be umbers wth the property r( j p+)+l or r( j p + ) + l < 0, for each j, j p. The the followg s vald a l (A r a r pr l+ A l pr )p Equalty holds f ad oly f a = a 2 = = a. ( r ) p. (3) Proof Suppose that umbers l,r ad p satsfy the codtos of Theorem. The, for each a, 0 a A, =,2,..., the followg equalty s vald a l (A r a r = )p (p )! j=p ( j) p j p+)+l ar( Ar( j+),

3 Lower Bouds of the Krchhoff ad Degree Krchhoff Idces 27 where ( j) p = j( j ) ( j p + 2), ( j) 0 = j. Summg the above equalty o, =,2,...,, we obta a l (A r a r )p = = (p )! (p )! j=p j=p ( j) p j p+)+l ar( Ar( j+) = ( j) p Ar( j+) r( j p+)+l a. Whe apply dscrete Jese equalty (see [9]) to the above equalty we obta a l (A r a r )p = (p )! j=p ) l ( ( j) p ) a r( j p+)+l = Ar( j+) ( A (A r A r r ) p = pr l+ A l pr ( r ) p The followg theorem ca be smlarly proved. Theorem 2 Suppose that real umbers a,a 2,...,a ad umbers l,r ad p satsfy the codtos of Theorem. The a l (A r a r ) p a l (A r a r + ( )pr l+ (A a ) l pr )p (( ) r ) p. (4) Equalty holds f ad oly f a 2 = a 3 = = a. We ow obta a lower bouds for some graph varats. Theorem 3 Let G be a udrected coected graph wth, 3, vertces ad m edges. If for umbers l ad p, p, hold j p + l 0 for each j, j p, the (2m ( 2)µ ) l µ p ( ) p l+ (2m) l p (5) Equalty holds f ad oly f G = K. Proof. For :=, r =, a = A µ, =,2,...,, equalty (3) becomes (A µ ) l µ p ( )p l+ A l p ( 2) p (6)

4 28 I. Ž. Mlovaovć etall. Based o the equalty a = A µ, =,2,..., ad (), we obta that A = 2m 2. By substtutg A = 2m 2 (6) we arrve at (5). Havg md that equalty (3), for :=, occurs f ad oly f a = a 2 = = a, equalty (5) holds f ad oly f G = K. Corollary Let G be udrected coected graph wth, 3, vertces ad m edges. The, for each p, p the followg s vald Equalty holds f ad oly f G = K. ( )p+ µ p (2m) p. Corollary 2 Let G be udrected coected graph wth, 3, vertces ad m edges. The ( )2 ( )2 K f (G) 2m Equalty holds f ad oly f G = K. Theorem 4 Let G be udrected coected graph wth, 3, vertces ad m edges. The, for each p, p ( )p+ ( µ ) p ( 2) 2m p. Equalty holds f ad oly f G = K. Corollary 3 Let G be udrected coected graph wth, 3, vertces ad m edges. The, for each p, p µ p p + ( 2)p+ (2m ) p. Equalty holds f ad oly f G = K. Corollary 4 Let G be udrected coected graph wth, 3, vertces ad m edges. The ( 2)2 K f (G) + 2m. Equalty holds f ad oly f G = K.

5 Lower Bouds of the Krchhoff ad Degree Krchhoff Idces 29 Corollary 5 [22] Let G be udrected coected graph wth, 3, vertces ad m edges. The ( 2)p+ µ p + ( + ) p (2m ) p. Equalty holds f ad oly f G = K. Corollary 6 [23] Let G be udrected coected graph wth, 3, vertces ad m edges. The K f (G) ( 2) m. Equalty holds f ad oly f G = K. Corollary 7 Let G be udrected coected graph wth, 3, vertces ad m edges. The K f (G) 2 ( ) 2m. 2m Equalty holds f ad oly f G = K. We ow obta equaltes that a specal case assg lower boud for graph varat DK f (G). Theorem 5 Let G be udrected coected graph wth, 3, vertces ad m edges. The, for each l ad p, p, that obey equalty j p + l 0, for each j, j p, the followg s vald Equalty holds f ad oly f G = K. ( ( 2)ρ ) l ρ p ( ) l p+ l p. (7) Proof. For :=, r =, a = A ρ, =,2,...,, equalty (3) trasforms to (A ρ ) l ρ p ( )p l+ A l p ( 2) p. Accordg to the equalty a = A ρ ad (2) we have that A = 2. Substtutg A = 2 the above equalty we arrve at (7). For :=, equalty (3) holds f ad oly f a = a 2 = = a, so the equalty (7) holds f ad oly f ρ = ρ 2 = = ρ,.e. whe G = K.

6 30 I. Ž. Mlovaovć etall. Corollary 8 Let G be udrected coected graph wth, 3, vertces ad m edges. The, for each p, p 2m 2m( )p+ ρ p p. Equalty holds f ad oly f G = K. Corollary 9 [2] Let G be udrected coected graph wth, 3, vertces ad m edges. The 2m( )2 DK f (G). Equalty holds f ad oly f G = K. Theorem 6 Let G be udrected coected graph wth, 3, vertces ad m edges. The, for each p, p 2m 2m( )p+ ( ρ ) p p ( 2) p Equalty holds f ad oly f G = K. Ackowledgemet Ths research was supported by the Serba Mstry of Educato ad Scece, uder grat TR3202. Refereces [] C. ARAUZ, The Krchhoff dexes of some composte etworks, Dscrete Appl. Math., 60 (202), [2] M. BIACHI, A. CORNARO, J. L. PALACIS, A. TORRIERO, Boud for Krchhoff dex va majorzato techques, J. Math. Chem., 5 (203), [3] S. B. BOZKURT, D. BOZKURT, O the sum of powers of ormalzed Laplaca egevalues of graph, MATCH Commu. Math. Comput. Chem., 68 (202), [4] M. CAVERS, S. FALLAT, S. KIRKLAND, O the ormalzed Laplaca eergy ad the geeral Radć dex R of graphs, L. Algebra Appl., 433 (200), [5] H. CHEN, F. ZHANG, Resstace dstace ad the ormalzed Laplaca spectrum, Ds. Appl. Math., 55 (2007), [6] F. R. K. CHUNG, Spectral graph theory, Am. Math. Soc., Provdece, 997. [7] S. M. CIOABÂ, Sums of powers of the degrees of a graph, Dscrete Math., 306 (6) (2006),

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