New Bounds for the Resolvent Energy of Graphs

Transcription

1 SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER A: APPL MATH INFORM AND MECH vol 9, 2 207), 87-9 New Bouds for the Resolvet Eergy of Graphs E H Zogić, E R Glogić Abstract: The resolvet eergy of a graph G of order is defied as ERG) = λ i), where λ λ 2 λ are the eigevalues of G Lower ad upper bouds for the resolvet eergy of a graph, which deped o some of the parameters,λ,λ,detr A )) = are obtaied Keywords: resolvet eergy, graph, iequalities Itroductio λ i, Let M be a square matrix of order The resolvet matrix, R M z), of matrix M is defied as [9] R M z) = zi M), where I is the uit matrix of order ad z a complex variable As easily see, R M z) is also a matrix of order, that exists for all values of z except whe z coicides with a eigevalue of M Let G be the simple graph, A its adjacecy matrix ad λ λ 2 λ eigevalues of A The resolvet matrix, R A z), is defied as ad its eigevalues are R A z) = zi A), z λ i,i =,2,, Bearig i mid that λ i for all i =,2,,, [4], we could choose z = Now we have that λ i,i =,2,, are the eigevalues of matrix R A ) = I A) ad detr A )) = λ i Mauscript received Jue 7, 207 ; accepted October 2, 207 E H Zogić ad E R Glogić are with the State Uiversity of Novi Pazar, Departmet of Mathematical Scieces, Novi Pazar, Serbia 87

2 New Bouds for the Resolvet Eergy of Graphs 88 Let G be a graph o vertices, >, with eigevalues λ,λ 2,,λ Its resolvet eergy is [7] ERG) = λ i Some remarkable properties of ERG) were revealed i [7] There are results about defiig ERG) via spectral momets ad characteristic polyomial of graphs, ad some bouds for the ERG) i terms of parameters,m, 0, where m is the umber of edges ad 0 is a ullity of the graph Additioal properties of ERG) ca be also foud i the recet papers [, 5, 6, ] I this paper, we obtaied some ew lower ad upper bouds for the resolvet eergy of a graph i terms of,λ,λ ad detr A )) 2 Some commo iequalities ad prelimiary lemmas Now, we itroduce some commo iequalities which we eed for our proofs i the sectio of mai results Lemma 2 [8] Let a i,r,r R, 0 < r a i R, i =,, The a 2 i a i 2 R r)2 ) Lemma 22 [8] Let a i,,r,r R, 0 < r a i R, i =,,, a i + rr Lemma 23 [0] Let a i R +,i =,, The ) ) a i + a i = The a i r + R 2) ai Lemma 24 [8] Let a i,,r,r R, 0 < r a i R, i =,,, ) a i + ) a i 3) = The a i R r a i 4 r + 4) R Lemma 25 [3] Let 0 < a a i a k a, p, p 2,, p be positive real umbers such that p + p p = ad Q i = p + p 2 + +,R k = p k + p k+ + p The p a + p 2 a p a p a + p 2 a p a Q ir k a k a i a i a k Q i a i + R k a k ), 5)

3 New Bouds for the Resolvet Eergy of Graphs 89 with equality for a = a 2 = = a i,a k = a k+ = = a,a i+ = a i+2 = = a k = Q i a i +R k a k Q i +R k Lemma 26 [2] Let p, p 2,, p be o-egative real umbers ad a,a 2,,a ad b,b 2,,b real umbers with the properties 0 < r a i R < + ad 0 < r 2 b i R 2 < + for each i =,2,, Further, let S be a subset of I = {,2,,} which miimizes the expressio 2 The a i b i a i b i R r )R 2 r 2 ) ) 6) Lemma 27 [4] A graph has oe eigevalue if ad oly if it is totally discoected A graph has two distict eigevalues λ > λ 2 with multiplicities m ad m 2 if ad oly if it cosists of m complete graphs of order λ + I that case, λ 2 = ad m 2 = m λ Lemma 28 [8] Let a = a i ),b = b i ),c = c i ) be three sequeces of real umbers of the same mootoicity ad p = ) sequece of real umber The a i b i c i a i b i c i 7) If a = a i ) ad b = b i ) are oppositely ordered, the the sese of ieqality 7) reverses 3 Mai results We represet some ew lower ad upper bouds for the resolvet eergy of graphs Theorem 3 Let G be a graph o vertices with eigevalues λ λ 2 λ The ERG) detr A ))) + 2 ) λ λ 8) λ ) λ ) Equality is attaied if ad oly if G = K Proof Let s cosider iequalities ) ad 3), where we could choose r = λ,r = λ,a i = λi,i =,,, to obtai λ i λi λi 2 λ + ) λ i λ 9) ) λi 0)

4 New Bouds for the Resolvet Eergy of Graphs 90 have From the defiitio of the resolvet eergy of graph, ERG) = ER λi + 2 λ ) 2 λ λ i, ad by 9), we ER + ) λ i = ER + )detr A ))) + 2 ) + 2 λ λ λ ) λ ) λ λ λ ) λ ), where i the secod iequality we used 0) Now, it follows that ERG) detr A ))) + 2 ) λ λ λ ) λ ) If G = K the ERK ) = ad i 8) equality holds If equality holds i 8) the equality is attaied i 9) i 0), from wich follows that λ = λ 2 = = λ By the Lemma 27 it follows that G = K Theorem 32 Let G be a graph o vertices with eigevalues λ λ 2 λ The ERG) 2 + λ ) λ ) ) 2 λ λ ) Proof Usig Chebyshev iequality for 3 sequeces 7) ad usig iequality 2), for a i = λ i, =,i =,,, r = λ,r = λ we obtai a lower boud ) Theorem 33 Let G be a graph o vertices with eigevalues λ λ 2 λ The ERG) + λ λ λ ) λ )2 λ λ ) 2) Proof For a i = λ i, =,i =,,, Q i = R k =, the iequality 5) trasforms ito ERG) + λ λ λ ) λ )2 λ λ ) Theorem 34 Let G be a graph o vertices with eigevalues λ λ 2 λ The ERG) 4 2 λ λ λ ) λ ) 3) Proof Usig =,a i = λ i,i =,,,r = λ,r = λ, i the 4) we obtai the upper boud 3)

5 New Bouds for the Resolvet Eergy of Graphs 9 Theorem 35 Let G be a graph o vertices with eigevalues λ λ 2 λ The λ λ λ λ 2 α) + detr A)) ERG) λ λ λ λ 2 α) + detr A )) Proof The proof follows from the iequality 6) for a i = b i = λi,i =,,,r = r 2 = λ,r = R 2 = λ Refereces [] L E Allem, J Capaverde, V Trevisa, I Gutma, E Zogić, E Glogić, Resolvet Eergy of Uicyclic, Bicyclic ad Tricyclic Graphs, MATCH Commu Math Comput Chem ) [2] Adrica, D, Badea, C, Grüs iequality for positive liear fuctioals, Period Math Hugar, 9 988), o 2, [3] V Cirtoaje, The Best Lower Boud Depeded o Two Fixed Variables for Jeses Iequality with Ordered Variables, Joural of Iequalities ad Applicatios, ), -2 [4] D Cvetković, M Doob, H Sachs, Spectra of Graphs - Theory ad Applicatio, Academic Press, New York, 980 [5] Z Du, Asymptotic Expressios for Resolvet Eergies of Paths ad Cycles, MATCH Commu Math Comput Chem ) [6] M Ghebleh, A Kaso, D Stevaović, O Trees with Smallest Resolvet Eergy, MATCH Commu Math Comput Chem ) [7] I Gutma, B Furtula, E Zogić, E Glogić Resolvet eergy of graphs, MATCH Commu Math Comput Chem [8] D S Mitriović, P Vasić, Aalytic Iequalities, Spriger, Berli, 970 [9] TS Shores, Liear Algebra ad Matrix Aalysis, Spriger, New York, 2007 [0] B Zhou, I Gutma, T Aleksić A ote o Laplacia eergy of graphs, MATCH Commu Math Comput Chem ) [] Z Zhu, Some extremal properties of the resolvet eergy, Estrada ad resolvet Estrada idices of graphs, Joural of Mathematical Aalysis ad Applicatios )