Resolvent Estrada Index of Cycles and Paths

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1 SCIENTIFIC PUBLICATIONS OF THE STATE UNIVERSITY OF NOVI PAZAR SER. A: APPL. MATH. INFORM. AND MECH. vol. 8, 1 (216), 1-1. Resolvet Estrada Idex of Cycles ad Paths Bo Deg, Shouzhog Wag, Iva Gutma Abstract: Let G be a simple graph of order. The resolvet Estrada idex of G is defied as EE r = ( 1, 1 λ i 1) where λ1,λ 2,...,λ are the eigevalues of G. Formulas for computig EE r of the cycle C ad the path P are derived. The precisio of these approximatios are show to be excellet. We also examie the differece ad relatios betwee the Estrada idex ad the resolvet Estrada idex of C ad P. Keywords: Resolvet Estrada idex, Estrada idex, spectrum (of graph), cycle, path. 1 Itroductio Let G be a simple graph of order with vertex set {v 1,v 2,...,v }. The adjacecy matrix of G, is the square matrix A = (a i j ) of order, i which a i j = 1 if the vertices v i ad v j are adjacet, ad a i j = otherwise. The characteristic polyomial ϕ(g,λ) of the graph G is the polyomial of degree, defied as det(λ I A), where I is the uit matrix of order. Sice A is real ad symmetric, all its eigevalues λ 1,λ 2,...,λ are real. These form the spectrum of the graph G. For a iteger k, the k-th spectral momet of G is defied as M k = M k (G) = I this paper, we are cocered with two simple ad frequetly ecoutered graphs the cycle C ad the path P. Recall that C is the coected graph of order i which all vertices have degree 2. The path P is the tree (= coected acyclic graph) of order i Mauscript received October 21, 215; accepted Jauary 14, 216. Bo Deg is with Guagdog Uiversity of Petrochemical Techology, Mao Mig, Guagdog 525, Chia, ad Ceter for Combiatorics ad LPMC-TJKLC, Nakai Uiversity, Tiaji 352, Chia; Shouzhog Wag is with the College of Sciece, Guagdog Uiversity of Petrochemical Techology, Maomig, Guagdog, 525, P. R. Chia; I. Gutma is with the Faculty of Sciece, Uiversity of Kragujevac, Serbia, ad the State Uiversity of Novi Pazar, Serbia. λ k i. 1

2 2 Bo Deg, Shouzhog Wag, Iva Gutma which exactly two vertices have degree 1 (whereas all other vertices have degree 2). The spectra of C ad P are well kow [4] (see below). I 2, Eresto Estrada itroduced a structural ivariat based o the spectral momets, defied as [6] EE = EE(G) = which evetually was amed Estrada idex [14]. Applicatios of the Estrada idex rage from the descriptio of foldig of protei molecules [6, 7] to measurig the cetrality of complex (commuicatio, social, metabolic, etc.) etworks [9, 8]. This graph spectrum based ivariat was also subject of extesive mathematical studies, for review see [1]. Recetly, Estrada ad Higham [8] modified the quatity occurrig o the right had side of Eq. (1), ad cosidered the spectrum based ivariat EE r = EE r (G) = k= k= M k k! M k (1) ( 1) k. (2) Bearig i mid that λ i < 1 holds for for all eigevalues of all graphs of order, except i the case of the -vertex complete graph [4], the right had side summatio i Eq. (2) is coverget. Thus, EE r is well defied for all graphs, except for the complete graphs. It is easy to verify that EE r = 1 = 1 λ i ( 1 λ ) 1 i. (3) 1 The expressio o the right had side of Eq. (3) idicates that EE r is a resolvet operator based quatity [1], i view of which EE r has bee amed resolvet Estrada idex. Although EE r has may properties aalogous to those of EE, the two idices are distict i essece [8, 3]. I what follows, we will show additioal evidece to support this view. At the preset momet, there are oly a few mathematical ad computatioal studies of the resolvet Estrada idex [3, 2, 11, 12]. Some of the basic properties of EE r have bee established, but umerous ope problems (may stated i form of cojectures [12]) await to be solved i the future. The preset work is aimed at cotributig towards partially fillig this gap. Cycles ad paths play a importat role i researchig the resolvet Estrada idex. I particular, we have: Theorem 1 [11] Amog all coected graphs of order ( 1), the path P has miimal resolvet Estrada idex. I order to fid the graph with secod miimal EE r -value, the tree P 1 ( j) had to be cosidered [5, 11], where P ( 1)( j) is obtaied by attachig a pedet vertex at positio j of the path P 1. The it follows:

3 Resolvet Estrada Idex of Cycles ad Paths 3 Theorem 2 [11] Amog all coected graphs of order ( 4), the tree P 1 (2) has the secod miimal resolvet Estrada idex. For a complete proof of Theorem 2, it was ecessary to show that [11] EE r (P 1 (2)) < EE r (C ). The latter iequality holds because C possesses may more self-returig walks tha P 1 (2) ad some of the odd spectral momets of C are greater tha zero if is odd. Similarly, whe j 3, the coditio EE r (P 1 ( j)) < EE r (C ) is ecessary for establishig the j-th miimal resolvet Estrada idex amog all coected graphs of order [11]. I [12], based o extesive computer work, it has bee show that the cycle C has smallest EE r -value amog all coected uicyclic graphs of order ( 3). Motivated by the above results, we ow offer a simple method for computig the resolvet Estrada idices of P ad C. Before presetig it, we outlie the aalogous formulas for the ordiary Estrada idex. 2 Estrada idices of cycles ad paths Graovac ad oe of the preset authors [13] showed that the Estrada idices of the cycles ad paths ca be approximated as EE(C ) I (4) ad EE(P ) ( 1)I cosh(2) (5) where I = k 1/(k!) 2 = Hece, i the case of cycles ad paths, EE ca be calculated easily. I Fig. 1 are preseted the EE(C )- ad EE(P )-values for 1 < 16. As it ca be see, except for the first few values of, the plots of EE(C ) ad EE(P ) are practically parallel, ad EE(C ) is always greater tha EE(P ). This is i full agreemet with Eqs. (4) ad (5).

4 4 Bo Deg, Shouzhog Wag, Iva Gutma 35 C 3 P 25 2 EE Fig. 1. EE(C ) (upper) ad EE(P ) (lower), plotted versus. 3 Resolvet Estrada idex of cycles It is well kow [4] that the spectrum of the cycle C cosists of the umbers (2iπ/), i = 1,2,...,. I view of this, EE r (C ) = = ( 1 1 ) 2iπ 1 1 2iπ 1 2iπ 4 1 2iπ 2 1 2iπ 5 1 2iπ 3 1 2iπ 6 1 Oe ca see that the agles 2iπ/ uiformly cover the iterval [,2π] whe i = 1,2,...,.

5 Resolvet Estrada Idex of Cycles ad Paths 5 Thus, the followig itegral approximatio is applicable: EE r (C ) = 2iπ 1 2iπ 2 1 2iπ 3 1 2iπ 4 1 2iπ 5 1 2iπ 6 1 2π 2π xdx 2π( 1) 2π( 1) 2 (x) 2 dx 2π 2π( 1) 3 (x) 3 2π dx 2π( 1) 4 (x) 4 dx 2π 2π( 1) 5 (x) 5 2π dx 2π( 1) 6 (x) 6 dx. By direct computatio, we obtai EE r (C ) 2 ( 1) 2 6 ( 1) 4 2 ( 1) 6. (6) The precisio of the approximate expressio (6) ca be see from the data i Table 1. Except for the first few values of, the resolvet Estrada idices of cycles are excelletly reproduced by Eq. (6). Thus, its accuracy is o two, four, ad six decimal places for, respectively, 5, 7, ad 11. This is more tha sufficiet for ay stadard applicatio of EE r (C ). 4 Resolvet Estrada idex of paths I a similar way as i the case of C, yet somewhat more complicated, we get a approximate expressio for EE r (P ). The spectrum of P cosists of the umbers [iπ/(

6 6 Bo Deg, Shouzhog Wag, Iva Gutma EE r (C ) EE r (C ) approx EE r (P ) EE r (P ) approx Table 1. Exact ad approximate values of resolvet Estrada idices of C ad P. 1)], i = 1,2,..., [4]. I view of this, ( EE r (P ) = = 1 1 iπ 1 1 ) 1 iπ 1 1 iπ iπ iπ iπ iπ

7 Resolvet Estrada Idex of Cycles ad Paths 7 Sice the agles iπ/(1) do ot cover the etire iterval [,π], the missig ear zero ad ear-π cotributios eed to be compesated whe applyig a itegral approximatio. Bearig this i mid, we ca proceed as follows. Let m > be a iteger. The, iπ m 1 = Note that 1 2 i= ( iπ 1 1 ) m [ 2 m 2 m ] 1 1 iπ m π 2π( 1) m (x) m dx 1 π 2π( 1) m (x) m dx 1 [ 2 m 2 m ] = 1 π π( 1) m (x) m dx 1 [ 2 m 2 m ] m 1 iπ 1 is approximately equal to whe m is odd. Therefore, EE r (P ) 1 π π( 1) 2 (x) 2 dx 22 ( 1) 2 1 π π( 1) 4 (x) 4 dx 2 4 ( 1) 4 1 π π( 1) 6 (x) 6 dx 26 ( 1) 6. This fially yields EE r (P ) ( 1) 2 ( 1) 4 ( 1) 6. (7) I Table 1 are also give the exact ad approximate values of EE r (P ). We see that the precisio of the approximatio (7) is remarkably good. Comparig the values of EE r (C ) ad EE r (P ), it becomes evidet that, except for the first few values of, these two resolvet Estrada idices are almost equal. The very same coclusio is obtaied by comparig the expressios (6) ad (7).

8 8 Bo Deg, Shouzhog Wag, Iva Gutma 5 Comparig Estrada ad resolvet Estrada idices I this sectio, we examie the relatios ad differece betwee the Estrada idex ad the resolvet Estrada idex i the case of C ad P. As see from Eqs. (1) ad (2), the resolvet Estrada idex depeds o all eigevalues or all spectral momets of the uderlyig graph. Yet, as show by our formulas (6) ad (7), i the case of cycles ad paths this idex ca be calculated (approximately, but with very high precisio) from just the umber of vertices,. I Fig. 2 are show EE r (C ) ad EE r (P ) for 2 < < 16. We see that, except for the first few values of, EE r (C ) ad EE r (P ) are almost equal. This fidig is i full agreemet with the approximate expressios deduced i Sectios 3 ad C 14 P 12 1 EE r Fig. 2. EE r (C ) ad EE r (P ), plotted versus. By comparig Figs. 1 ad 2, a remarkable differece is evisaged betwee the Estrada idex ad the resolvet Estrada idex i the case of cycles ad paths. The reaso for this differece ca be explaied as follows. From the expressios (4) ad (5), it is immediately see that, except for the first few values of, EE(C ) ad EE(P ) are liear fuctios of. Due to the same slope I of Eqs. (4) ad (5), the two lies show i Fig. 1 are parallel. The differece betwee the two lies is give by the term cosh(2) I, which is a fixed costat, ad is approximately equal to O the other had, the expressios (6) ad (7) for the resolvet Estrada idices of C ad P are o-liear fuctios of, although their deviatio from liearity rapidly vaishes

9 Resolvet Estrada Idex of Cycles ad Paths 9 with icreasig. The differece betwee (6) ad (7) is 2 ( 1) 2 1 ( 1) 4 44 ( 1) 6 which rapidly teds to zero for. This implies that the two lies preseted i Fig. 2 gradually ted to become liear ad ted to coicide as icreases. That this happes very fast is see from Fig. 2 ad the data give i Table 1. By comparig expressios (4) ad (6) ad bearig i mid that I , we easily see that EE r (C ) < EE(C ) (8) holds provided 4. Ideed, if 4, the ad the by (4) ad (6), 2 ( 1) 2 6 ( 1) 4 2 ( 1) ( 1) 2 6 ( 1) 4 2 ( 1) < I implyig the iequality (8). I a aalogous maer, for 3, we get EE r (P ) < EE(P ). (9) Iequalities (8) ad (9) shed some light o the relatios betwee the Estrada ad the resolvet Estrada idices. It remais a challege for the future to ivestigate whether the iequality EE r (G) < EE(G) holds for other graphs G, or perhaps for all graphs. If ot, the it would be iterestig to fid graphs for which EE r (G) = EE(G). Ackowledgemets. This research was supported by the Sciece ad Techology Project of Maomig City o , ad by the Itroductio of Talet Project o Refereces [1] M. Bezi, P. Boito, Quadrature rule based bouds for fuctios of adjacecy matrices, Li. Algebra Appl., Vol. 433, (21),

10 1 Bo Deg, Shouzhog Wag, Iva Gutma [2] X. Che, J. Qia, Boudig the resolvet Estrada idex of a graph, J. Math. Study, Vol. 45, 2 (212), [3] X. Che, J. Qia, O resolvet Estrada idex, MATCH Commu. Math. Comput. Chem., Vol. 73, 1 (215), [4] D. Cvetković, M. Doob, H. Sachs, Spectra of Graphs Theory ad Applicatio, Academic Press, New York, 198. [5] H. Deg, A ote o the Estrada idex of trees, MATCH Commu. Math. Comput. Chem., Vol. 62, 3 (29), [6] E. Estrada, Characterizatio of 3D molecular structure, Chem. Phys. Lett., Vol. 319, (2), [7] E. Estrada, Characterizatio of the foldig degree of proteis, Bioiformatics, Vol. 18, (22), [8] E. Estrada, D. J. Higham, Network properties revealed through matrix fuctios, SIAM Rev., Vol. 52 (21) [9] E. Estrada, J. A. Rodríguez Velázquez, Subgraph cetrality i complex etworks, Phys. Rev. E, Vol. 71, (25), [1] I. Gutma, H. Deg, S. Radeković, The Estrada idex: A updated survey, i: D. Cvetković, I. Gutma (Eds.), Selected Topics o Applicatios of Graph Spectra, Math. Ist., Beograd, 211, pp [11] I. Gutma B. Furtula, X. Che, J. Qia, Graphs with smallest resolvet Estrada idices, MATCH Commu. Math. Comput. Chem., Vol. 73, 1 (215), [12] I. Gutma, B. Furtula, X. Che, J. Qia, Resolvet Estrada idex computatioal ad mathematical studies, MATCH Commu. Math. Comput. Chem., Vol. 74, 3 (215), [13] I. Gutma, A. Graovac, Estrada idices of cycles ad paths, Chem. Phys. Lett., Vol. 463, (27), [14] J. A. de la Peña, I. Gutma, J. Rada, Estimatig the Estrada idex, Li. Algebra Appl., Vol. 427, (27), 7 76.

New Bounds for the Resolvent Energy of Graphs

New Bounds for the Resolvent Energy of Graphs 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