Randić Energy and Randić Estrada Index of a Graph

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1 EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, ISSN SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL TURKEY Randć Energy and Randć Estrada Index of a Graph Ş. B u r c u B o z k u r t,durmuşbozkurt Department of Mathematcs, Scence Faculty, Selçuk Unversty, 42075, Campus, Konya, Turkey Abstract. Let G be a smple connected graph wth n vertces and let d be the degree of ts -th vertex. The Randć matrxofg s the square matrx of order n whose, j -entry s equal to / fthe -th and j-th vertex of G are adjacent, and zero otherwse. The Randć egenvaluesaretheegenvalues of the Randć matrx. The Randć energy s the sum of the absolute values of the Randć egenvalues. In ths paper, we ntroduce a new ndex of the graph G whch s called Randć Estrada ndex. In addton, we obtan lower and upper bounds for the Randć energy and the Randć Estrada ndex of G Mathematcs Subject Classfcatons: 05C50,5A8 Key Words and Phrases: Randć Matrx, Randć Egenvalue, Randć Energy, Randć Estrada Index. Introducton Let G be a smple connected graph wth n vertces and m edges. Throughout ths paper, such a graph wll be refered to as connected (n, m)-graph. Let V (G) = v, v 2,...,v n be the vertex set of G. Ifanytwovetcesv and v j of G are adjacent, then we use the notaton v v j. For v V (G), the degree of the vertex v, denoted by d, s the number of the vertces adjacent to v. Let A(G) be the (0, )-adjacency matrx of G and λ, λ 2,...,λ n be ts egenvalues. These are sad to be egenvalues of the graph G and to form ts spectrum [3]. TheRandć matrx of G s the n n matrx R = R (G)= R j as the followng R j = d d j, v v j 0, otherwse Correspondng author. Emal addresses: (Ş. Bozkurt), (D. Bozkurt) 88 c 202 EJPAM All rghts reserved.

2 Ş. B o z k u r t a n d D. B o z k u r t / Eur. J. Pure Appl. Math, 5 (202), The Randćegenvaluesρ, ρ 2,...,ρ n of the graph G are the egenvalues of ts Randć matrx. Snce A(G) and R (G) are real symmetrc matrces, ther egenvalues are real numbers. So we can order them so that λ λ 2... λ n and ρ ρ 2... ρ n. The energy of the graph G s defned n [,2,3] as: E = E (G)= The Randć energyofthegraphg s defned n [,2] as: RE = RE (G)= The Estrada ndex of the graph G s defned n [7,8,9,0] as: λ. () EE = EE(G)= Denotng by M k = M k (G) the k-th moment of the graph G M k = M k (G)= Recallng the power seres expanson of e x,wehave EE = k=0 ρ. (2) e λ (3) k λ. M k k!. (4) It s well known that [3] M k s equal to the number of closed walks of length k n the graph G. EstradandexofgraphshasanmportantrolenChemstryand Physcs. For more nformaton we refer to the reader [7,8,9,0]. In addton, there exst a vast lterature that studes Estrada ndex and ts bounds. For detaled nformaton we may also refer to the reader [4,5,6,4]. Now we ntroduce the Randć Estrada ndex of the graph G. Defnton. If G s a connected (n, m)-graph, then the Randć Estrada ndex of G, denoted by REE (G), s equal to REE = REE (G)= e ρ. (5) where ρ, ρ 2,...,ρ n are the Randć egenvalues of G. Let k N k = N k (G)= ρ.

3 Ş. B o z k u r t a n d D. B o z k u r t / Eur. J. Pure Appl. Math, 5 (202), Recallng the power seres expanson of e x we have another expresson of Randć Estrada ndex as the followng N k REE (G)= k!. (6) In ths paper, we obtan lower and upper bounds for the Randć energyandtherandć Estrada ndex of G. Frstly,wegvesomedefntonsandlemmaswhchwllbeneeded then. Defnton 2. [2] Let G be a graph wth vertex set V (G) = v, v 2,...,v n and Randć matrx R. Then the Randć degree of v, denoted by R s gven by R = R j. j= Defnton 3. [2] Let G be a graph wth vertex set V (G) = v, v 2,...,v n and Randć matrx R. Let the Randć degree sequence be R, R 2,...,R n. Then for each =, 2,..., n the sequence L (), L (2),...,,... s defned as follows: Fx α, let and for each p 2, let L () k=0 = R α = L (p ). j d d j Defnton 4. [5] Let G be a graph wth Randć matrx R. Then the Randć ndex of G, denoted by R (G) s gven by R (G)= R. 2 Lemma. [] Let G be a graph wth n vertces and Randć matrx R. Then tr(r)= ρ = 0 and tr R 2 = ρ 2 = 2. Lemma 2. [2] Let G be a connected graph α be a real number and p be an nteger. Then Sp+ ρ. S p where S p = f 2. Moreover, the equalty holds for partcular values of α and p f and only L (p+) = L(p+) 2 2 = = L(p+) n. n

4 Ş. B o z k u r t a n d D. B o z k u r t / Eur. J. Pure Appl. Math, 5 (202), Lemma 3. [2] A smple connected graph G has two dstnct Randć egenvalues f and only f G s complete. 2. Bounds for the Randć Energy of a Graph In ths secton, we obtan lower and upper bounds for the Randć energyofconnected (n, m)-graph G. Let N and M be two postve ntegers. We frst consder the followng auxlary quantty Q as N Q = Q (G)= q (7) where q, =, 2,..., N are some numbers whch some how can be computed from the graph G. Forwhchweonlyneedtoknowthattheysatsfythecondtons q 0, for all =, 2,..., N and N 2 q = 2M (8) or, the condtons (7), (8) and P = P (G)= N q (9) f all the condtons (7)-(9) are taken nto account then [] 2MN (N ) D Q 2MN D (0) where D = 2M NP 2/N. () For the graph energy (namely by settng nto (0) and () N = n, M = m and P = det A ), ths yelds [] 2m + n (n ) det A 2/n E (G) 2m (n )+n det A 2/n. The Randć energy-counterpart of the estmates(0)and()s obtaned as the followng result. Theorem. Let G be a connected (n, m)-graph and be the absolute value of the determnant of the Randć matrx R. Then RE (G) 2 + n (n ) 2/n (2)

5 Ş. B o z k u r t a n d D. B o z k u r t / Eur. J. Pure Appl. Math, 5 (202), and RE (G) 2 (n ) + n 2/n. (3) Proof. The result s easly obtaned usng the estmates (0), () and Lemma. In [] the followng result for RE (G) was obtaned RE (G) 2n (4) Remark. The upper bound (3) s sharper than the upper bound (4). Usng arthmetcgeometrc mean nequalty, we obtan 2 n 2/n and consderng the upper bound (3) we arrve at whch s the upper bound (4). RE (G) 2n 3. Bounds for the Randć Estrada Index of a Graph In ths secton, we consder the Randć Estrada ndex of connected(n, m)-graph G. We also adapt the some results n [4] and [4] on the Randć Estradandextogvelowerand upper bounds for t. Theorem 2. Let G be a connected (n, m)-graph. Then REE(G) e Sp+ where α s a real number, p s an nteger and S p = (5) f and only f G s the complete graph K n. Sp + n e Sp+ n Sp. (5) 2. Moreover, the equalty holds n Proof. Startng wth the equaton (5) and usng arthmetc-geometrc mean nequalty, we obtan REE(G) = e ρ + e ρ e ρ n

6 Ş. B o z k u r t a n d D. B o z k u r t / Eur. J. Pure Appl. Math, 5 (202), e ρ +(n ) Now we consder the followng functon for x > 0. We have n =2 e ρ n = e ρ +(n ) e ρ n,snce f (x)=e x + n e x n f (x)=e x e x n > 0 (6) ρ = 0. (7) for x > 0. It s easy to see that f s an ncreasng functon for x > 0. From the equaton (7) and Lemma 2, we obtan Sp+ REE(G) e Sp + n. (8) n Sp e Now we assume that the equalty holds n (5). Then all nequaltes n the above argument must be equaltes. From (8) we have Sp+ Sp+ ρ = S p whch mples L(p+) = L(p+) 2 2 = = L(p+) n n.from(6)andarthmetc-geometrcmeannequalty we get ρ 2 = ρ 3 = = ρ n.thereforeg has exactly two dstnct Randć egenvalues, by Lemma 3, G s the complete graph K n. Conversely, one can easly see that the equalty holds n (5) forthecompletegraphk n. Ths completes the proof of theorem. Now we gve a result whch states a lower bound for the Randć Estradandexnvolvng Randć ndex. Corollary. Let G be a connected (n, m)-graph. Then REE(G) e 2R(G) n + n e 2R(G) n(n ) where R (G) denotes the Randć ndex of the graph G. Moreover the equalty holds n (9) f and only f G s the complete graph K n. Proof. In [2], theauthorsshowedthatthefolllowngnequalty(seetheorem 4) Sp+ R 2 2R (G) ρ n n S p (9) (20)

7 Ş. B o z k u r t a n d D. B o z k u r t / Eur. J. Pure Appl. Math, 5 (202), where S p = 2.CombnngTheorem2and(20)wegetthenequalty(9). Also, the equalty holds n (9) f and only f G s the complete graph K n. Theorem 3. Let G be a connected (n, m)-graph.then the Randć Estrada ndex REE (G) and the Randć energy RE (G) satsfy the followng nequalty 2 RE (G)(e )+n n + REE (G) n + e RE(G) 2. (2) where n + denotes the number of postve Randć egenvalues of G. Moreover, the equalty holds on both sdes of (2) f and only f G = K. Proof. Lower bound: Snce e x ex,equaltyholdsfandonlyfx = ande x + x, equalty holds f and only f x = 0, we get REE (G) = e ρ = e ρ + ρ >0 ρ 0 eρ + + ρ ρ >0 ρ 0 e ρ = e ρ + ρ ρ n+ + n n+ + ρn ρ n = (e ) ρ + ρ ρ n+ + n n+ + = 2 RE (G)(e )+n n +. Upper bound: Snce f (x)=e x monotoncally ncreases n the nterval (, + ), weget REE (G)= e ρ n n + + n + e ρ ρ Therefore REE (G) = n n + + = n + k n + k n + k ρ k 0 n + k! k! ρ k n + k! k ρ = n + e RE(G) 2. It s easy to see that the equalty holds on both sdes of (2) f and only f RE (G)=0. Snce G s a connected graph ths only happens n the case of G = K.

8 REFERENCES Concludng Remarks In ths paper, the Randć Estradandexofagraphsntroduced. AlsotheRandćenergy and the Randć Estrada ndex are studed. In secton 2,a sharper upper bound and a new lower bound for the Randć energyareobtaned. Insecton3,someboundsfortheRandć Estrada ndex nvolvng Randć ndex,randć energyandsome other graph nvarants are also put forward. ACKNOWLEDGEMENTS: The authors thank the referees for ther helpful suggestonscon- cernng the presentaton of ths paper. References [] Ş Bozkurt, A Güngör, I Gutman and A Çevk. Randć Matrx and Randć Energy. MATCH Commun. Math. Comput. Chem [2] Ş B o z k u r t, A G ü n g ö r, a n d I G u t m a n. R a n d ćspectralradusandrandćenergy.match Commun. Math. Comput. Chem [3] D Cvetkovć, M Doob and H Sachs. Spectra of Graphs-Theory and Applcaton (Thrd ed. Johann Ambrosus Bart Verlag, Hedelberg, Lepzg) [4] KDasandSLee.OntheEstradandexconjecture.Lnear Algebra Appl [5] JDelaPeña, IGutmanandJRada. EstmatngtheEstradandex. Lnear Algebra Appl [6] HDeng, SRadenkovćandIGutman. The Estrada ndex n: D. Cvetkovc, I. Gutman (Eds.) Applcatons of Graph Spectra (Math. Inst., Belgrade) [7] EEstrada.Characterzatonof3Dmolecularstructure.Chem. Phys. Lett [8] EEstrada.Characterzatonofthefoldngdegreeofprotens. Bonformatcs [9] EEstradaandJRodríguez-Velázguez.Subgraphcentraltyncomplexnetworks.Phys. Rev. E [0] EEstrada,JRodríguez-VelázguezadnMRandć. Atomc Branchng n molecules. Int. J.Quantum Chem [] IGutman.Boundsfortotalπ-electron energy. Chem. Phys. Lett [2] IGutman.Totalπ-electron energy of benezod hydrocarbons. Topcs Curr. Chem

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