Effective Resistances for Ladder Like Chains

Transcription

1 Effetive Resistanes for Ladder Like Chains Carmona, A., Eninas, A.M., Mitjana, M. May 9, 04 Abstrat Here we onsider a lass of generalized linear hains; that is, the ladder like hains as a perturbation of a n path by adding onseutive weighted edges between opposite verties. This lass of hains in partiular inludes a big family of networks that goes from the yle, uniyle hains up to ladder networks. In this paper, we obtain the Green funtion, the effetive resistane and the Kirhhoff index of those ladder like hains as funtion of the the Green funtion, the effetive resistane and the Kirhhoff index of a path. Departament de Matemàtia Apliada III, UPC-BarelonaTeh Departament de Matemàtia Apliada I, UPC-BarelonaTeh

2 INTRODUCTION The Kirhhoff Index was introdued in Chemistry as a better alternative to other parameters used for disriminating among different moleules with similar shapes and strutures; see?. Sine then, a new line of researh with a onsiderable amount of prodution has been developed and the Kirhhoff Index has been omputed for some lasses of graphs with symmetries; see for instane 8? and the referenes therein. This index is defined as the sum of all effetive resistanes between any pair of verties of the network and it is also known as the Total Resistane;?. To find the Kirhhoff index of a general network has a high degree of omputational omplexity. Hene, it is of interest to find losed formulae for the effetive resistane and the Kirhhoff index. Some works have been published in this diretion, for networks suh that yles, hexagonal hain, distane regular graphs, see.one an also raise the problem of omputing the Kirhhoff index of omposite networks in terms of fators,. In this work we deal with the omputation of Green funtion, effetive resistane and Kirhhoff Index of generalized linear hain. These networks an be obtained from a n path by adding edges between opposite verties. Hene, they are a perturbation of the path and we an apply the result obtained in 5 to obtain the desired parameters. Let Γ = V, E, be a network; this is a simple and finite onneted graph with vertex set V = {,,..., n} and edge set E, where eah edge i, j has been assigned a ondutane ij > 0. Moreover, when i, j / E we define ij = 0, in partiular ii = 0 for any i =,..., n. The weighted degree of vertex i is defined as δ i = n j= ij. The ombinatorial Laplaian of Γ is the matrix L, whose entries are L ij = ij for all i j and L ii = δ i. Therefore, for eah vetor u R n and for eah i =,..., n n Lu i = δ i u i ij = j= n ij u i u j. j= It is well known that Lu = 0 iff u = ae, a R and e is the all vetor. Moreover, the multipliity of 0 as eigenvalue of L is equal to the number of onneted omponents of Γ. As Γ is onneted, the projetor onto the trivial eigenspae is J/n, where J is the all matrix,

3 onsequently L J/n is non singular and we define the Green matrix of Γ as G = L J/n J/n. In other words, G is the Moore Penrose inverse of the Laplaian matrix L. For any pair i, j V, the effetive resistane between i and j is defined as R ij = u i u j, where u R n is any solution of the linear system Lu = e i e j, where e i denotes the ith unit vetor with in the ith position and 0 elsewhere. Note that R ij does not depend on the hosen solution and in addition, the following equality holds, R ij = G ii G jj G ij. It is well known that, for any i, j, k V the triangular inequality R ij R ik R kj is an equality iff k separates verties i and j. The Kirhhoff index of Γ is the value, k = n n R ij = n G ii. i,j= i= In order to define the objets we are going to work with, we first onsider a fixed a path P on n verties, labelled as V = {,..., n}. The lass of generalized linear hains supported by the path P, denoted by L n, onsists of all onneted networks whose ondutane satisfies that i = i i > 0 for i =,..., n, a i = i n i 0 for any i =,..., n and ij = 0 otherwise. We define the link number of Γ as s = {i =,..., n : a i > 0} whih orresponds with the numbers of holes or quadrangles. So, the link number of Γ L n equals 0 iff a = = a n = 0; that is, iff the underlying graph of Γ is nothing but the path P. On the other hand, if the link number of Γ is positive there exist indexes i < < i s n suh that a ik > 0 when k =,..., s, whereas a j = 0 otherwise, see Figure. Generalized linear hains with link number s = are uniyle. In partiular, the n yle orresponds to the ase a > 0 and a j = 0, j =,..., n. A generalized linear hain whose link number equals n is alled a linear hain or ladder in the Graph Theory framework. Let G and R be the Green funtion and the effetive resistane of the path P. Sine eah interior vertex in a path is a ut vertex, we get R ij = R min{k,i} min{k,j} R max{k,i} max{k,j}, i, j, k =,..., n. 3 3

4 i k n n a i a ik a is n n n n n n i k n Figure : A Generalized Linear Chain The authors proved in 3 that for any i, j =,..., n, the Green funtion of a path is G ij = min{i,j} k n n k max{i,j} kn k, 4n k= k=max{i,j} k=min{i,j} where we use the usual onvention that empty sums are defined as zero. effetive resistane and the Kirhhoff index of the path are, Therefore, the R ij = max{i,j} k=min{i,j} n, i, j =,..., n and k = k= kn k. Moreover, for a path with onstant ondutanes, the expression of the Green funtion is G ij = n 4n 3 ii n jj n n i j n and hene, k = n i j 3 4n and R ij = for any i, j =,..., n. Given Γ L n, we denote its Green funtion as G Γ. If Γ has positive link number s and {i j } s j= is its link sequene, then the ombinatorial Laplaian of Γ appears as the ombinatorial Laplaian of the weighted path perturbed by adding for all j =,..., s, an edge with ondutane a ij between opposite verties i j and n i j 4,5. Sine we interpret a generalized linear hain as a perturbation of the path by adding weighted edges between opposite verties, we use 5 Theorem. to obtain the Green funtion, the effetive resistanes and the Kirhhoff index of suh a hain. To this end, we onsider the s s matrix Λ with entries Λ jk = a ij a ik R max{ij,i k } n max{i j,i k } and we take into aount that I Λ is non singular beause it is positive definite. Let M be its inverse. 4

5 For any j =,..., n, we define the vetor v j whose omponents are v jk = aik Rn ik j R ik j, k =,..., s, 4 and also the vetor u j = Mv j. Moreover, we onsider the vetor r = n Aording to the previous notation, the Green funtion, the effetive resistane and the Kirhhoff index of a generalized linear hain, are given by the following result. Theorem For any i, j =,..., n, we get n j= v j. G Γ ij = G ij Mr v i, r v j and R Γ ij = R ij u i u j, v i v j. In partiular, the Kirhhoff index of the generalized linear hain is given by k Γ = k 4n Mr, r n n j= u j, v j. Identity 3 allows us to give nie expressions for vetors v j and r. To do this, it is useful to define for any h =,..., n, the funtion φ h : {,..., n} {h,..., n h} given by φ h j = h, j h, j, h j n h, n h, n h j n. Clearly, φ h is nondereasing and moreover, given j, k =,..., s, we have that Λ jk = a ij a ik R φik i j φ ik n i j = a ij a ik R φij i k φ ij n i k. Lemma For any k =,..., s, we have v j,k = v j,k = v i,k v j,k = aik aik aik R ik n i k R ik φ ik j, j n, R φik j n ik R ik n ik, n j n, R φ ik i φ ik j, i j n. 5

6 aik s In partiular, v n i = v i = R i k n i k, v j = v i and v n j = v n i for k= n i aik k j n any j i. Moreover, r k =, whih in turns implies n j=i j k r k v j,k = aik n Uniyle linear hains n i k m=i k m n n i k m=φ ik j, j n. In this setion we obtain the Green funtion, the effetive resistane and the Kirhhoff index for uniyle linear hains; that is, for those generalized linear hains whose link number equals one. Therefore if i = h, then we add an edge with ondutane n = a h > 0 between verties h and n h. Sine s =, the omputation of M and u j, j =,..., n, is straightforward. Thus, I Λ = n R h n h = n n j=h. 5 Moreover, for any i, j =,..., n, we have the following useful version of Identity 3, j R ij = R min{h,i} min{h,j} R max{n h,i} max{n h,j} R φh i φ h j. 6 Proposition 3 For any i, j =,..., n, we get that G Γ ij = min{i,j} k n n k 4n R Γ ij = 4n k= n min{h,max{i,j}} k=min{h,i,j} n k=max{i,j} k n max{n h,i,j} k=φ h i k=max{n h,min{i,j}} φ h max{i,j} k=φ h min{i,j} n max{i,j} k=min{i,j} kn k φ h min{i,j} k n k=φ h j k=φ h max{i,j}. 6

7 In partiular, k Γ = n k= kn k n k n Proof. For any i, j =,..., n, we have m=k nh G Γ ij = G ij M r v i r v j and R Γ ij = R ij Mv i v j.. The expression for the Green funtion is a onsequene of the last identity in Lemma, whereas the expression for the effetive resistane appears as a onsequene of the mentioned Lemma and Identity 6. Finally, sine k Γ = n On the other hand, n m n j= sine m=h n j= k Γ = k n n j= m=φ h j n G Γ jj, we have n j= m=φ h j m=φ h j = n m=h n 4n j= = 4n h n = h m=h = h j= m 4n m=φ h j m n m=h m=h m=h m 4n j=h m=j m=h and hene, the expression for the Kirhhoff index follows. 7 j=h m=φ h j n j= m m=φ h j m=j = m=h j=h m=j m,. n m=h m,

8 Corollary 4 The Kirhhoff index of an uniyle hain with onstant ondutanes a and is k Γ = n 4n an h nn h 4hh. 3 an h Next, we partiularize the above theorem to h = that orresponds to the n yle. Although the ase of yles with onstant weight and ondutanes is well known, see for instane 6, as far as authors knowledge, this is the first time that the orthogonal Green funtion for a weighted yle is obtained. Corollary 5 The Green funtion and the effetive resistane for the weighted n yle are given by G Γ ij = 4n min{i,j} k= n 4n k= n Rij Γ = k= k n k= max{i,j} n k=max{i,j} k=min{i,j} Moreover, the Kirhhoff index is given by k Γ = n k= n kn k k= k n n k n k=i min{i,j} k= n k= max{i,j} k=min{i,j} n k n k= k n n kn k n k=j k=max{i,j} In partiular, if = for all k =,..., n and n = a, we get,. n n j= k=j. G Γ ij = n 4n 3 ii n jj n n i j n R Γ ij = a 4 an n i n j, i j an a n i j, and hene, k Γ = n4n an 3 an. 8

9 The last expression, when a =, oinides with the one obtained in 6. To end this setion we rise the problem of optimizing the Kirhhoff index of an uniyle linear hain. Firstly, we assume a =, then for h n, k Γ h = n8h3 h n 8hn 4n 3 h hn 8n h n 6a n h 7 and its derivative with respet to h is k Γ h = n6h3 36n h 4n h 4n 3 n 5n 4 6a n h. Let ψh = 6h 3 36n h 4n h 4n 3 n 5n 4, as ψ h = 48h n h n vanishes only at n / in the interval, n. Therefore, ψ vanishes at most one in, n/ and one at most in n/, n. Moreover, ψ < 0 on n/, n and ψn > 0, then ψ is positive on n/, n. On the other hand, ψ n = 3an < 0 4 and ψ n = 9 n 4 4 an > 0, so the k Γ h has a minimum for h, n. 4 4 Taking into aount that h Z, the minimum value of the Kirhhoff index for uniyle n linear hains is attained for h =. Finally, the maximum is attained for h = n sine 4 k Γ k Γ n. Let us now assume a, more preisely a = λ with λ. Then, for h n k Γ h, λ = n 8λh 3 λn h λ4n 6n h 4λn n λn 4n. 3hλ λn λ Let k Γ h, λ be its derivative with respet to h. It an be easily proved that for λ >, k Γ h, λ vanishes at the interval n, n 4 4 n and the minimum of the funtion is attained again for h =. On the other hand, for 4 λ < this is a <, there is a zero of k Γ h, λ in the interval n λ, n. Consequently, 4 4 when the ondutane a is very small ompared with, the Kirhhoff index of the uniyle hain is minimum for h =. In any ase, the Kirhhoff index of the uniyle hain with ondutanes a and, reahes its maximum for h = n. 9

10 Ladder like hains A ladder like hain is a generalized linear hain obtained by adding s onseutive edges, s n h, from vertex h with the same ondutane a > 0, to the path with onstant ondutane > 0, see Figure, so i k = h k and a ik = a > 0, k =,..., s. In partiular, when s = n the orresponding generalized linear hain is nothing else but a ladder network. In this setion, we aim to ompute the Green funtion, the effetive resistane and the Kirhhoff index for the ladder like hain. h h h s n a a a a n n h n h s n Figure : A ladder like hain Assoiated with the ladder like hain, we onsider q = a and we define the following auxiliary funtion in terms of Chebyshev polynomials n h s Vk q U k q, k 0, Q k q = n h s k 3 k 0, where V k q = U k q U k q and U k q is the k th Chebyshev polynomials of seond kind, see 7 and Annex A. Observe that Q 0 q is defined unambiguously, sine V 0 q = U 0 q =. Moreover, it is lear that {Q k q} k 0 is a Chebyshev sequene. aount that V k = and U k = k for any k Z, then Q k q = n h s V k U k, for any k 0, whih implies that {Q k q} k 0 is also a Chebyshev sequene. In addition, taking into From Lemma we get v n j = v j whih leads to u n j = u j, for any j =,..., n and moreover, r = 0. Therefore, for ladder like hains, Theorem reads as follows. 0

11 Corollary 6 For any i, j =,..., n, we get G Γ ij = G Γ n i n j = G ij u i, v j, G Γ i n j = G Γ n i j = G i n j u i, v j, Rij Γ = Rn i Γ n j = R ij u i, v i u j, v j u i, v j, Ri Γ n j = Rn i Γ j = R i n j u i, v i u j, v j u i, v j. Moreover, k Γ = n n 3 4n 4n u j, v j. j= Notie that in order to ompute the Green funtion and the effetive resistane for a ladder like hain, it suffies to obtain the values u i, v j for any i, j =,..., n. Therefore, the key is to ompute M. Applying the results of 5 Proposition.5 we get the following expression for M. Lemma 7 If M = b ij, then av min{i,j} qq s max{i,j} q b ij = δ ij V s q a n s h U s q Notie that when s =, the above formula gives b = with the inverse of 5 for onstant ondutanes., that oinides an h Next results are devoted to obtain the vetors v j, u j, j =,..., n and their inner produt. Proposition 8 It is satisfied that v j = v h and hene u j = u h, j =,..., h. Moreover, for any j =,..., n and any m =,..., s, we get a v j,m = n φ hm j, a u j,m = V min{φh j h,m }qq s max{φh j h,m }q V s q a n s h U s q. Proof. Given j =,..., n and m =,..., s, from Lemma we have v j,m = a n i m φ im j i m = a n φ im j, and the expression for v j,m follows bearing in mind that i m = h m.

12 Sine u j m =,..., s, we have a u j,m = a a α V m q = Mv j, if we onsider α = V s q a n s h U s q, then for n φ hm j a α s k=m a Q s mq m k= Q s k q n φ hk j V k q n φ hk j The result follows after arefully onsidering eah of the ases j h; h j h s ; h s j n and applying Lemma 3 in Annex A. Corollary 9 Given i, j n, then. u i, v j = a n i n j U s q 4 V s q a n s h U s q when h s i, j n, whereas u i, v j = n φh max{i, j} V φ h min{i,j} hqq s h φh max{i,j}q 4 V s q a n s h, U s q otherwise. One we have obtained the inner produt, the expression for the Green funtion of a ladder like hain is straightforward. Next we ompute the effetive resistane of a ladder like hain aording to the position in the path of the involved verties. Corollary 0 For any i, j =,..., n, we get i j V s q a n s h i j U s q Rij Γ = Rn i Γ n j = V s q a n s h U s q, Ri Γ n j = Rn i Γ j = n i j V s q a i j s h U s q V s q a n s h U s q. for h s i, j n and Rij Γ = Rn i Γ i j n j = V min{i,j} h q Q sh min{i,j} q Q sh max{i,j} q 4 V s q a n s h U s q Q sh max{i,j} q V max{i,j} h q V min{i,j} h q 4 V s q a n s h U s q

13 Ri Γ n j = Rn i Γ n max{i, j} j = V min{i,j} h q Q sh min{i,j} q Q sh max{i,j} q 4 V s q a n s h U s q Q sh max{i,j} q V max{i,j} h q V min{i,j} h q 4 V s q a n s h U s q for h i, j h s. Proposition The Kirhhoff index of the ladder like hain is where k Γ = n4n 3 n h s n h ss n h Q s q fn, h, su s q gn, h, su s q Vs q a n h s U s q, fn, h, s = gn, h, s = Proof. First, we have n u j, v j = a 4α U s q j=hs in addition, u j, v j = 4 hs j= Moreover, hs n h s as s a an h s n h s n h s 3 sn h s. a n j=hs 3 n j = a α U s q n h s n h s n h s 3, hs j= n φh j V φh j hqq s h φh jq V s q a n s h. U s q n φh j = h n h hs n j j= j=h = h s n h ss, 3

14 whereas taking into aount the last identity in Lemma 4, hs j= Thus, V φh j hqq s h φh jq = h Q s q k Γ = n4n 3 = n4n 3 na n j=hs = n4n 3 hs 4n u j, v j 4n n j= hs j= = h Q s q s V j qq s j q j= sn h s a U s q n h s as s U s q, a n u j, v j j=hs n φh j n j Us q 4 V s q a n s h U s q n h s n h ss V φh j hqq s h φh jq V s q a n s h U s q nh Q s q Vs q a n s h U s q snn h s U s q a V s q a n s h U s q n n h s as s U s q a V s q a n s h U s q nan h s n h s n h s 3 U s q 3 V s q a n s h U s q. result. For the standard ladder; that is, when s = n and hene h =, we have the following Corollary The Kirhhoff index of the standard ladder is k Γ = nn 3 In partiular, when a =, then n n U n q an a U n q a V n q au n q. k Γ = nn 3 n 4 n k=0 sin kπ n.

15 Proof. The first Identity follows substituting s = n and h = in Proposition. On the other hand, when a =, then q = and n n U n nu n 3U n = nu n U n = n n k= os kπ n sine { os kπ n } n k= are the zeroes of the Chebyshev polynomial U n, see 7. In the Chemistry ommunity, standard ladders are known as linear polyomino hains. Then, the last formula oinides with that obtained in 8 Theorem 4. for a linear polyomino hain with n squares. Annex A In this setion we write the results related with the Chebyshev sequenes that we need aross the paper. The following Lemma shows a useful property for the sum of Chebyshev polynomials, see for instane 7. Lemma 3 If {P k } k=0 is a Chebyshev sequene, given Sk = αk β, where α, β R, and r, t N suh that t r then, r k=t SkP k q = Sr P r q P r q St P t q P t q α P t q P r q. a From the expression for produts of Chebyshev polynomials, see 7 Chapter, we dedue the following equalities. Lemma 4 For any m s we have aq s m qu m q V m q Q s m q Q s m q = V s q a n s h U s q. Moreover, if for any k Z, T k and W k denote the k th Chebyshev polynomial of first and fourth kind, respetively; that is W k q = U k q U k q, then s V j qq s j q = j= s n h s T s q sw s q n h s U s q, a 5

16 whih is equivalent to s sn h s V j qq s j q = q j= Proof. s V j qq s j q = j= s j= = s j= n h s qs s U s q U s q q n U j q U j q h s Us j q U s j q U s j q n U j q U j q h s 3 Us j q n h s U s j q = n h s 3 s j= n h s s n h s 3 = q n h s j= q n h s s = q n h s q q U s j qu j q U s j qu j q U s j qu j q U s j qu j q st s q st s q s T s j q st s q j= s T s j q st s q j= T s q T s q T s q s T s j q j= s T s j q j= s j= T s j q s T s j q j= s q s T s j q j= s T s j q j= n h s sq = q T s q n h s q U s q s T s q T s q q s j= T s q T s q T s j q = sn h s q n h s qs s U s q U s q q taking into aount s s T s l q = T s l q = U s q T s q = qu s q, l= l= s T s l q = U s q. l= 6

17 CONCLUSIONS We have obtained the Green funtion, the effetive resistane, and the Kirhhoff index of a lass of generalized linear hains that inludes yles, uniyle hains and ladder hains also known as polyomino hains in the Chemistry ommunity. The starting point is a path with arbitrary ondutanes on their edges, then we interpret eah generalized hain as a onvenient perturbation of the mentioned path. Therefore, we obtain the expressions of the Green funtion, the effetive resistane, and the Kirhhoff index as funtion of its orresponding in the path. In partiular we obtain, as far as we know for the first time, the Green funtion of a weighted yle. We expliitly give the Kirhhoff index for uniyle hains with two different ondutanes and we disuss when the Kirhhoff index is optimum aording to the size of the yle. The last setion is devoted to the study of ladder like hains again as a perturbation of the path. For them we have also find the desired results. In order to ahieve the last goal, we have had to deal with Chebyshev polynomials tools that are inluded in Annex A. ACKNOWLEDGMENTS This work has been partly supported by the Spanish Researh Counil Comisión Interministerial de Cienia y Tenología, under projets MTM C0-0 and MTM C0-0. 7

18 Referenes. E. Bendito, A. Carmona, A. M. Eninas, and J. M. Gesto, Linear Algebra Appl. 430, , ISSN , URL E. Bendito, A. Carmona, A. M. Eninas, J. M. Gesto, and M. Mitjana, Linear Algebra Appl. 43, 78 00, ISSN , URL 3. E. Bendito, A. Carmona, A. Eninas, and M. Mitjana, Linear Algebra Appl. 436, 090 0, ISSN , URL 4. A. Carmona, A. M. Eninas, and M. Mitjana, Linear Algebra Appl. 44, 5 04, ISSN , URL 5. A. Carmona, A. Eninas, and M. Mitjana, Linear Algebra and its Appliations pp. 04, ISSN , URL 6. R. B. Ellis, Ph.D. thesis, University of California, San Diego J. Mason and D. Handsomb, Chebyshev Polynomials Chapman & Hall/CRC, Y. Yang and H. Zhang, Int. J. Quantum Chem. 08, , URL 8

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