On the energy of symmetric matrices and Coulson s integral formula

Transcription

1 Rvista Colombiana d Matmáticas Volumn , páginas On th nrgy of symmtric matrics and Coulson s intgral formula Sobr la nrgía d matrics simétricas y la fórmula intgral d Coulson J. A. d la Pña 1, J. Rada 2, 1 Cntro d Invstigación n Matmáticas, A.C., Guanajuato, México 2 Univrsidad d Antioquia, Mdllín, Colombia Abstract. W dfin th outr nrgy of a ral symmtric matrix M as E out M = λ i λ M for th ignvalus λ 1,..., λ n of M and thir arithmtic man λm. W discuss th proprtis of th outr nrgy in contrast to th innr nrgy dfind as E innm = n λ i. W prov that E inn is th maximum among th nrgy functions : Sn R and E out among functions f M λm1 n, whr f is an nrgy function. W prov a variant of th Coulson intgral formula for th outr nrgy. Ky words and phrass. Total π-lctron nrgy, Enrgy of a symmtric matrix, Bounds for nrgy, Coulson s intgral formula Mathmatics Subjct Classification. 05C50. Rsumn. Dfinimos la nrgía xtrior d una matriz simétrica ral M como E out M = λi λ M dond λ 1,..., λ n son los autovalors d M y λm s su mdia aritmética. Discutimos las propidads d la nrgía xtrior n contrast con la nrgía 175

2 176 J. A. DE LA PEÑA & J. RADA intrior dfinida como E innm = n λ i. Dmostramos qu E inn s máxima ntr todas las funcions d nrgía : Sn R y E out ntr todas las funcions f M λm1 n, dond f s una función d nrgía. Dmostramos una variant d la fórmula intgral d Coulson para la nrgía xtrior. Palabras y frass clav. Enrgía π-lctrón total, Enrgía d una matriz simétrica, Cotas para la nrgía, Fórmula intgral d Coulson. 1. Introduction Rsarch on nrgy of a graph gos back to th work of Erich Hückl on th approximat solution of th Schrödingr quation of crtain organic molculs. Gnralizing from facts obsrvd in this molcular thory, Gutman introducd in [7] th dfinition of th nrgy of a graph G as EG = λ i whr n is th numbr of vrtics and λ 1,..., λ n ar th ignvalus of th adjacncy matrix AG = a ij, dfind as a ij = 1 if thr is an dg btwn i and j and 0 othrwis. Dtails on th dvlopmnt of th mathmatical concpt and its associatd chmistry applications can b sn in th rcnt book [14]. W dnot by ϕ G x or simply ϕx whn no confusion ariss th charactristic polynomial of th graph G dfind as ϕx = dt x1 n AG which is a monic polynomial of dgr n whos roots ar th ignvalus λ 1,..., λ n of th adjacncy matrix AG. In th thory of graph nrgy a prominnt rol is playd by th Coulson intgral formula EG = 1 π n ixϕ ix ϕix dx whr ϕ x = d dx ϕx is th first drivativ of ϕx. A drivation of th intgral formula as wll as svral of its chmical applications can b sn in [14] s also [10]. In 2006 Gutman and Zhou [11] introducd a nw nrgy basd on th Laplacian matrix of a graph G, dfind as L = D A, whr D = diag d 1,..., d n is th diagonal matrix of vrtx dgrs and A is th adjacncy matrix of G. Not that if µ 1,..., µ n ar th ignvalus of L thn n µ i = 2m, whr m is th numbr of dgs of G, and thus is trivial. With th intntion to conciv a graph-nrgy-lik quantity that prsrvs th main faturs of th original Volumn 50, Númro 2, Año 2016

3 ON THE ENERGY OF SYMMETRIC MATRICES AND COULSON S INTEGRAL FORMULA177 graph nrgy, Gutman and Zhou dfind th Laplacian nrgy of a graph G with n vrtics and m dgs as LE G = n µi 2m n. Latr Consonni and Todschini [6] usd th xprssion E M = n ξi ξ, whr ξ 1,..., ξ n ar th ignvalus of a molcular matrix and ξ is thir arithmtic man, for dsigning quantitativ structur-proprty rlations for a varity of physical-chmical proprtis of a numbr of classs of organic compounds. Not that in th cas of th Laplacian matrix L, th arithmtic man of its ignvalus is prcisly 2m n, so th Laplacian nrgy of a graph is a particular cas of this xprssion. Aftr this a larg numbr of nrgis appard in th mathmatical and mathmatico-chmical litratur, which wr basd on th ignvalus of matrics associatd to th graph. For xampl, th signlss Laplacian nrgy [1],[26], th normalizd Laplacian nrgy [5], th distanc nrgy [13],[25], also gnralizations to digraphs [18],[19],[20],[21],[22],[23],[24] and [2],[3],[4], among othrs. Motivatd by th analogous forms of various graph nrgis, Gutman [8] proposd an ultimat xtnsion of th graph-nrgy concpt as E X = x i x for numbrs x 1,..., x n and x thir arithmtic man valu. For mor dtails on nrgis of graphs and digraphs w rfr to [9],[15],[17]. In this papr w rfr to th nrgy concivd by Consonni and Todschini as to th outr nrgy of a ral symmtric matrix M as E out M = λi λ M for th ignvalus λ 1,..., λ n of M and thir arithmtic man λ M. In this papr w discuss th proprtis of this nrgy in contrast to th innr nrgy dfind as E inn M = λ i Obsrv that ths quantitis coincid in cas trm = 0, as happns for th adjacncy matrix of a graph. Lt Sn b th spac of ral symmtric n n-matrics. For M Sn w dnot by M j Sn 1 th matrix o btaind from M by dlting th j-th row and column, and M j 0 Sn is th matrix formd from M by rplacing th j-th row and column by zros. W shall say that f n : Sn R is an nrgy function if it satisfis th following conditions for vry n 1 : Rvista Colombiana d Matmáticas

4 178 J. A. DE LA PEÑA & J. RADA E0 f n is non-ngativ, unitarily invariant and f 1 1 = 1; E1 if M, N Sn thn f n M + N f n M + f n N and for r any scalar w hav f n rm = r f n M; E2 if M Sn has a j-th row and column of zros thn f n M = f n 1 M j, for all n 2. Thorm 1.1. Th innr nrgy E inn : Sn R is th maximal function among th nrgy functions f n : Sn R. W say that f n : Sn R is an affin-nrgy function if f n M = n M λm1n for an nrgy function n and th linar function 1 n tr = λ : Sn R. Thorm 1.2. Th outr nrgy E out : Sn R is th maximal function among th affin-nrgy functions f n : Sn R. W dnot ϕ M x or simply ϕx whn no confusion ariss th charactristic polynomial of th ral symmtric n n-matrix M dfind as ϕx = dt x1 n M. This a a monic polynomial of dgr n whos roots ar th ignvalus λ 1,..., λ n of M. Lt λ b th arithmtic man of th λ i. W considr th associatd affin polynomial ψ x = ϕ x λ whos roots ar of th form λ λ i. W prov th following variant of Coulson intgral formula: Thorm 1.3. Lt M b a ral symmtric n n-matrix. Th following holds: E out M = 1 π W mntion som applications of this formula. n ixψ ix ψix dx. 1 Exampls: 2. Th outr nrgy of a matrix a For 1 k n, dfin p k : Sn R by p k M = k λ i, whr th ignvalus of M ar ordrd as λ 1 M λ 2 M λ n M. p k is an nrgy function for ach 1 k n. In particular, E inn : Sn R is an nrgy function. Volumn 50, Númro 2, Año 2016

5 ON THE ENERGY OF SYMMETRIC MATRICES AND COULSON S INTEGRAL FORMULA179 b For M = m ij Sn dfin f n M = an nrgy function. 1 n m ij Thn fn is 2 Lt λ n,..., λ s+1 λ λ s,..., λ 1 b th ignvalus of a matrix M. W assum that th arithmtic man λ 0. Obsrv that E inn M nλ 0 and 1 + n s λ i E inn M = λ i + λ j λ i n sλ n Morovr E out M = λ i λ = whr im = 1 2s n λ j +n 2sλ i=s+1 λ i = λ j i=s+1 < 1 is th balanc indx of th matrix M. λ i +nimλ Proposition 2.1. Lt M = a ij b a matrix with ignvalus λ i with λ n,..., λ s+1 λ λ s,..., λ 1, whr 0 λ is th man valu of th λ i. Thn a 0 im < 1; b 1 2 E outm = s λj λ = n If morovr all λ i 0 thn c 2E inn M E out M = 2sMλ > 0. Proof. For a, obsrv that s > n 2 implis 0 λ j sλ = λj λ = λ λj. λ λj = λ+n sj λ n λ j < λ λ j and nλ < s + 1 λ nλ a contradiction showing that s n 2 and im 0. For b, obsrv that th inquality sλ + λ i λ i + λ j = nλ λ j + n sλ i=s+1 Rvista Colombiana d Matmáticas

6 180 J. A. DE LA PEÑA & J. RADA writtn as a + A B + b has th proprty that th sum of th first and last trms, a and b, rspctivly, quals th sum of th scond and third trms, A and B, rspctivly. That is, a + b = A + B. Thrfor, sinc 0 b A = B a = λ i This shows b. Thn λj λ = λ j + n 2s λ = E out M = i G λ λj = 1 2 E outm λi λ 0 For c, asum that 0 λ n,..., λ s+1 λ λ s,..., λ 1 such that E inn M = E inn M E out M = i=s+1 λ i + λ j = nλ 0 λ i n 2sλ = 1 2 E outm + smλ. Proposition 2.2. Lt M = a ij b a matrix with ignvalus λ i with λ n,..., λ s+1 λ λ s,..., λ 1, whr 0 λ is th man valu of th λ i. Thn th following assrtions ar quivalnt: a E inn M = E out M; b tr M = 0; c λ = 0. Proof. Only that a implis b is not clar. Assum that E inn M = E out M and lt Thn λ n λ t+1 0 λ t λ s+1 λ λ s,..., λ 1 λ i λ j +n 2sλ = i λi λ = Eout M = E inn M = t λ i j=t+1 λ j Volumn 50, Númro 2, Año 2016

7 ON THE ENERGY OF SYMMETRIC MATRICES AND COULSON S INTEGRAL FORMULA181 Hnc t 1 im λ i = n 2s λ = 2 λ i 2 t s λ. i=s+1 3 Proof of Thorm 1.1: First, obsrv that E inn is an nrgy function s Exampls. W show our rsult by induction, bing th cas n = 1 trivial. Lt f n : Sn R b an nrgy function and M Sn. Find a diagonal matrix D = diag λ 1,..., λ n quivalnt to M, thn f n M = f n D = f n = 1 n 1 = 1 n 1 1 n 1 D j 0 f n 1 D j 1 n 1 λ i = E inn M. j i 1 n 1 f n D j 0 E inn n 1 D j 4 Lt σ 1 M σ 2 M σ n M b th singular valus of M and σm th arithmtic man, that is, σ j M ar th squar roots of th ignvalus of MM. Obsrv that th symmtric matrix N = M λm1 n has singular valus λ i M λm. Thrfor E out M = λ i M λm = σ i N. In othr words, E out M is qual to Nikiforov s nrgy of N = M λm1 n s [16]. It follows from [12, Corollary 3.4.4] that E out satisfis E1. Proposition 2.3. Lt M Sn b a ral symmtric matrix. Thn E out M = 0 if and only if M = a1 n for som scalar a. Proof. Lt λ 1,..., λ n b th ignvalus of M. Obsrv that E out M = 0 implis that th man valu λ = λ i for all i and thrfor M = λ1 n. 5 Proof of Thorm 1.2: Not that E inn M λm1n = λ i λm = E out M so E out is a affin nrgy function. To chck maximality, lt f n : Sn R b any affin-nrgy function and M Sn. Thn f n M = n M λm1n for any nrgy function n and so by Thorm 1.1 f n M = n M λm1n Enn M λm1n = Eout M. Rvista Colombiana d Matmáticas

8 182 J. A. DE LA PEÑA & J. RADA 1 Invrs matrics 3. Som spcial classs of matrics Lt M b a ral symmtric n n-matrix. Dnot by ϕ M x th charactristic polynomial of dgr n whos roots ar th ignvalus λ 1,..., λ n of M. Lt λ b th arithmtic man of th λ i. Suppos M is invrtibl, thn all λ i 0 and λ 1 1,..., λ 1 n ar th ignvalus of M 1 with man valu λ 1 = 1 n λ 1 i = H λ 1,..., λ n 1 whr H λ 1,..., λ n is th harmonic man of th ignvalus. Th harmonic-arithmtic mans inquality stats that 1 H λ 1,..., λ n 1 λ = λ 1 λ. Mor prcis information may b obtaind as th following rsult shows: Proposition 3.1. Lt M b a non-singular ral symmtric n n-matrix. Thn a sm + s M 1 n; b λ 1 = λ 1 ; c E out M 1 = 1 λ 2 E out M. Proof. Writ µ i = λ 1 i and µ = λ 1. Thn λ λ i implis µ i λ 1 µ. Thrfor sm n s M 1 which yilds a. Calculat E out M 1 = 2 Morovr, i=sm 1 +1 µ µ i = 2 sm λi λ λ i λ 2λ sm 2 E out M 1 µ 2 E out M 1 1 µ 2 λ 2 E out M E out M λi λ = 1 λ 2 E outm which yilds qualitis b and c. 2 Exponntial matrics Lt M b a matrix with ignvalus λ n λ t+1 0 < λ t λ s+1 < λ λ s λ 1 Volumn 50, Númro 2, Año 2016

9 ON THE ENERGY OF SYMMETRIC MATRICES AND COULSON S INTEGRAL FORMULA183 and whos arithmtic man is λ 0. W considr th xponntial matrix M with ignvalus 0 < λn λt+1 λt λ1 and whos arithmtic man λ M > 0 satisfis λt+1 Proposition 3.2. With th notation abov w hav: λ M λ t. a λ λ M, with quality if and only if λ n = = λ 1 ; b s M sm and im i M ; c E out M 2 n t. Proof. Obsrv that th arithmtic man-gomtric man inquality yilds λ = n λi 1 n 1 n That is inquality a. For b, obsrv that λi = λ M λ n λ m+1 < 0 λ m λ s+1 < λ λ s λ t+1 < lnλ M λ t λ 1 and t = s M sm = s m. Hnc For c, considr E out M = 1 2 E out M t λ i λ M E out M + i=t+1 i=t+1 λ M λi + i=t+1 λ M λi λ M λi + m i=s+1 λ λi + i=s+1 i=m+1 λ λi λ λi th last inquality du to th fact that, for ach t + 1 i m, w hav λ λ i = c i 0 and λ i 0. Thn for k 1, Thrfor λ λi λ λi and λ k λ k i = λ i + c i k λ k i c k i m i=s+1 λ λi m s. Rvista Colombiana d Matmáticas

10 184 J. A. DE LA PEÑA & J. RADA Morovr, λ M λi λ t+1 λt s t i=t+1 i=t+1 and In conclusion λ λi n m i=m+1 E out M 2 n t A simpl xampl will illustrat th last statmnt. Considr th matrix 0 1 M = 1 0 with ignvalus λ 1 = 1 and λ 2 = 1. Thn E out M = 2. Morovr, m = t = s = 1. Th ignvalus of M ar and 1 with man µ = and whil th stimatd lowr bound is 2. E out M = On Coulson-lik formulas 1 Lt M b a ral symmtric n n-matrix. Dnot by ϕ M x or simply ϕ x whn no confusion ariss th charactristic polynomial dfind as ϕ x = dt x1 n M. This is a monic polynomial of dgr n whos roots ar th ignvalus λ 1,..., λ n of M. Lt λ b th arithmtic man of th λ i. W considr th associatd affin polynomial whos roots ar of th form λ λ i. ψ x = ϕ x λ, Lt ψ x = b n x n b n 1 x n n 1 b 1 x + 1 n b 0 and considr for 1 k n, k s k = λ λi Ths cofficints satisfy: Volumn 50, Númro 2, Año 2016

11 ON THE ENERGY OF SYMMETRIC MATRICES AND COULSON S INTEGRAL FORMULA185 i b n = 1; ii b 1 = 0 = s 1 ; iii Nwton s idntitis hold, namly: s 2 = b 1 s 1 2b 2, s 3 = b 2 s 1 b 1 s 2 + 3b 3, s 4 = b 3 s 1 b 2 s 2 + b 1 s 3 4b 4, tc... 2 Proof of Thorm 1.3: Lt λ j = λ j M b th ignvalus of M. Dfin f z = zϕ z λ ϕ z λ = zψ z ψ z Sinc thn ψ z ψ z = zψ z ψ z n = n 1 z λ + λ j λ j λ z λ + λ j Tak any closd contour Γ containing in its intrior xactly thos λ j λ 0. Th wll-known Cauchy formula in complx calculus yilds 1 fz n dz = λj λ = 1 2πi 2 E out Γ λ<λ j th last quality du to Proposition 2.1. Obsrv that th actual form of th contour Γ is unimportant. Thrfor w can inflat it as indicatd in [7] to obtain 1 π as dsird. n ixψ ix dx = 1 ψ ix πi Γ fz n dz = E out 3 Following [7], w stablish som dirct consquncs of th Coulson-lik rsult just provd. Corollary 4.1. Lt M b a ral symmtric n n-matrix. Dnot by ϕ x its charactristic polynomial and ψ x = ϕ x λ th associatd affin polynomial. Thn E out M = 1 n x d lnψ ix dx π dx Rvista Colombiana d Matmáticas

12 186 J. A. DE LA PEÑA & J. RADA Corollary 4.2. Lt M 1, M 2 b two ral symmtric n n-matrics. Dnot by ψ k x = ϕ k x λk th associatd afffin polynomials, k = 1, 2. Thn E out M 1 E out M 2 = 1 π ln ψ 1 ix ψ 2 ix dx. Applying th ordinary Coulson intgral formula and Thorm 1.3, w gt: Corollary 4.3. Lt M b a ral symmtric n n-matrix. Dnot by ϕ x its charactristic polynomial and ψ x = ϕ x λ th associatd affin polynomial. Thn E out M E inn M = 1 ln ψ ix π ϕ ix dx. Rfrncs [1] N. Abru, D. M. Cardoso, I. Gutman, E. A. Martins, and M. Robbiano, Bounds for th signlss Laplacian nrgy, Lin. Algbra Appl , [2] C. Adiga, R. Balakrishnan, and W. So, Th skw nrgy of a digraph, Lin. Algbra Appl , [3] C. Adiga and Z. Khoshbakht, On som inqualitis for th skw Laplacian nrgy of digraphs, J. Inqual. Pur Appl. Math , no. 3, Art. 80, 6p. [4] Q. Cai, X. Li, and J. Song, Nw skw laplacian nrgy of simpl digraphs, Trans. Comb , no. 1, [5] M. Cavrs, S. Fallat, and S. Kirkland, On th normalizd Laplacian nrgy and gnral Randić indx r 1 of graphs, Lin. Algbra Appl , [6] V. Consonni and R. Todschini, Nw spctral indx for molcul dscription, MATCH Commun. Math. Comput. Chm , [7] I. Gutman, Th nrgy of a graph, Br. Math.-Statist. Skt. Forschungszntrum Graz , [8], Bounds for all graph nrgis, Chm. Phys. Ltt , [9] I. Gutman and X. Li Eds, Enrgis of graphs - thory and applications, Univ. Kragujvac, Kragujvac, Volumn 50, Númro 2, Año 2016

13 ON THE ENERGY OF SYMMETRIC MATRICES AND COULSON S INTEGRAL FORMULA187 [10] I. Gutman and M. Matljvic, Not on th coulson intgral formula, J. Math. Chm , [11] I. Gutman and B. Zhou, Laplacian nrgy of a graph, Lin. Algbra Appl , [12] R. Horn and C. Johnson, Topics in matrix analysis, Cambridg Univrsity Prss, [13] G. Indulal, I. Gutman, and A. Vijayakumar, On distanc nrgy of graphs, MATCH Commun. Math. Comput. Chm , [14] X. Li, Y. Shi, and I. Gutman, Graph nrgy, Springr-Vrlag, Nw York, [15] M. Matljvic, V. Bozin, and I. Gutman, Enrgy of a polynomial and th intgral formula, J. Math. Chm , [16] V. Nikiforov, Th nrgy of graphs and matrics, J. Math. Anal. Appl , [17], Byond graph nrgy: Norms of graphs and matrics, Lin. Algbra Appl , [18] I. Pña and J. Rada, Enrgy of digraphs, Lin. Multilin. Algbra , [19] S. Pirzada and M. Bhat, Enrgy of signd digraphs, Discrt Appl. Math , [20] S. Pirzada, M. Bhat, I. Gutman, and J. Rada, On th nrgy of digraphs, Bull. IMVI , no. 1, [21] J. Rada, Th mccllland inquality for th nrgy of digraphs, Lin. Algbra Appl , [22], Lowr bound for th nrgy of digraphs, Lin. Algbra Appl , [23], Bounds for th nrgy of normal digraphs, Lin. Multilin. Algbra , [24] J. Rada, I. Gutman, and R. Cruz, Th nrgy of dirctd hxagonal systms, Lin. Algbra Appl , [25] H. S. Raman, I. Gutman, and D. S. Rvankar, Distanc quinrgtic graphs, MATCH Commun. Math. Comput. Chm , Rvista Colombiana d Matmáticas

14 188 J. A. DE LA PEÑA & J. RADA [26] W. So, M. Robbiano, N. M. M. d Abru, and I. Gutman, Applications of a thorm by ky fan in th thory of graph nrgy, Lin. Algbra Appl , Rcibido n julio d Acptado n sptimbr d 2016 Cntro d Invstigación n Matmáticas, A.C. Guanajuato, México -mail: jap@cimat.mx Instituto d Matmáticas Univrsidad d Antioquia Mdllín, Colombia -mail: pablo.rada@uda.du.co Volumn 50, Númro 2, Año 2016