A Simple Method for Computing Resistance Distance

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1 A Simple Method for Computing Resistance Distance Ravindra B Bapat, Ivan Gutman a,b, Wenjun Xiao b Indian Statistical Institute, New Delhi, 006, India a Faculty of Science, University of Kragujevac, P O Box 60, Kragujevac, Serbia Montenegro, b Xiamen University, P O Box 003, Xiamen 36005, P R China, Department of Computer Science, South China University of Technology, Guangzhou 5064, P R China Reprint requests to Prof I G; Fax: ; Z Naturforsch 58a, ; received August 2, 2003 The resistance distance r ij between two vertices v i v j of a connected, molecular graph G is equal to the effective resistance between the respective two points of an electrical network, constructed so as to correspond to G, such that the resistance of any edge is unity We show how r ij can be computed from the Laplacian matrix L of the graph G: Let Li Li, j be obtained from L by deleting its i-th row column, by deleting its i-th j-th rows columns, respectively Then r ij = detli, j/detli Key words: Resistance Distance; Laplacian Matrix; Kirchhoff Index; Molecular Graph Introduction When the structure of a molecule is represented by a metric topological space [ 5], then the distance between two vertices v i v j, denoted by d ij, is defined as the length = number of edges of a shortest path that connects v i v j in the corresponding molecular graph G The vertex-distance concept found numerous chemical applications; for details see the recent reviews [6 8] elsewhere [9 ] In order to examine other possible metrics in molecular graphs, Klein Rić [2] conceived the resistance distance between the vertices of a graph G, denoted by r ij, defined to be equal to the effective electrical resistance between the nodes i j of a network N corresponding to G, with unit resistors taken over any edge of N For acyclic graphs r ij = d ij, therefore the resistance distances are primarily of interest in the case of cycle-containing molecular graphs Resistance-distances molecular structure-descriptors based on them were much studied in the chemical literature [2 23] recently attracted the attention also of mathematicians [24, 25] In analogy to the classical Wiener index [7 ], one introduced [2] the sum of resistance distances of all pairs of vertices of a molecular graph, Kf = r ij, i< j a structure-descriptor that eventually was named [3] the Kirchhoff index Resistance distances are computed by methods of the theory of resistive electrical networks based on Ohm s Kirchhoff s laws; for details see [24] The stard method to compute r ij is via the Moore- Penrose generalized inverse L of the Laplacian matrix L of the underlying graph G: r ij =L ii +L jj L ij L ji 2 Equation 2 is stated already in [2] [4], but was, for sure, known much earlier In this work we communicate a novel, remarkably simple expression for r ij, stated in Theorem In order to be able to formulate our main result, we first specify our notation terminology remind the readers to some basic facts from Laplacian graph spectral theory 2 On Laplacian Spectral Theory Let G be a graph let its vertices be labeled by v,v 2,,v n The Laplacian matrix of G, denoted by L = LG, is a square matrix of order n whose i, j- entry is defined by L ij =, if i j the vertices v i v j are adjacent, L ij = 0, if i j the vertices v i v j are not adjacent, L ij = d i, if i = j, / 03 / $ 0600 c 2003 Verlag der Zeitschrift für Naturforschung, Tübingen

2 R B Bapat et al A Simple Method for Computing Resistance Distance 495 where d i is the degree = number of first neighbors of the vertex v i n Because L ij = n L ij = 0 for any graph G, i= j= detlg=0, i e, LG is singular The submatrix obtained from the Laplacian matrix L by deleting its i-th row the i-th column will be denoted by Li The submatrix obtained from the Laplacian matrix L by deleting its i-th j-th rows the i-th j-th columns will be denoted by Li, j, assuming that i j According to the famous matrix-tree theorem see for instance [26 29] for any graph G for any i =,2,,n, detli=tg, 3 where tg is the number of spanning trees of G The eigenvalues of the Laplacian matrix are referred to as the Laplacian eigenvalues form the Laplacian spectrum of the respective graph The Laplacian spectral theory is a well elaborated part of algebraic graph theory its details can be found in numerous reviews, for instance in [27, 30, 3] Concerning chemical applications of Laplacian spectra see [32, 33] We label the Laplacian eigenvalues of the graph G by i, i =, 2,,n, so that 2 n Then, n is always equal to zero, whereas n differs from zero if only if the underlying graph G is connected We are interested in molecular graphs, which necessarily are connected Therefore in what follows it will be understood that n 0 Of the many known relations between the structure of a graph its Laplacian spectrum [26 28, 30 33] we mention here only two: n i= i = ntg 4 n Kf = n 5 i i= 3 A Determinantal Formula for r ij Theorem Let G be a connected graph on n vertices, n 3, i j n Let Li Li, j be the above defined submatrices of the Laplacian matrix of the graph G Then detli, j r ij = detli 6 In view of 3, formula 6 can be written also as detli, j r ij = tg This remarkably simple expression for the resistance distance was discovered by one of the present authors [25] Here we demonstrate its validity in a manner different from that in [25] In order to prove Theorem we need some preparations To each vertex v i of the graph G associate a variable x i consider the auxiliary function f x,x 2,,x n =x k x l 2 k,l with the summation going over all pairs of adjacent vertices v k, v l Then for i j, { r ij = sup f x,x 2,,x n x i =,x j = 0, 0 x k, } k =,2,,n 7 Formula 7 is a result known in the theory of electrical networks cf Corollary 5 on p 30 in Chapt 9 of the book [24] Its immediate consequences are the following equations which hold for k i, j: f x k = 2 l Γ k x k x l =0, 8 where Γ k denotes the set of first neighbors of the vertex v k Because the vertices of the graph G are labeled in an arbitrary manner, without loss of generality we may restrict our considerations to the special case i = n j = n Then, however, the mathematical formalism of our analysis will become significantly simpler For the sake of simplicity, denote the submatrix Ln,n by L write the Laplacian matrix of G as L B LG= C D

3 496 R B Bapat et al A Simple Method for Computing Resistance Distance Then 8 can be written as L x t = b t, 9 where x =x,x 2,,x b =b,b 2,,b Because x n = x n = 0, if the vertices v n v k are adjacent then b k =, whereas otherwise b k = 0, k =,2,,n 2 Let A = a ij be the adjacency matrix of the graph G Then, in view of x n =, x n = 0, in addition to 9 we have xbx n,x n t =x n,x n C x t = a n,i x i 0 i= x n,x n Dx n,x n t = d n By combining 9 we obtain f x,x 2,,x n Because =x,x n,x n LGx,x n,x n t = xl x t +x n,x n C x t + xbx n,x n t +x n,x n Dx n,x n t = xb t 2 xb t = i= i= a n,i x i + d n a n,i x i = we finally arrive at the relation k Γ n f x,x 2,,x n =d n x k, k Γ n x k, 2 which holds under the extreme value condition x n =,x n =0 The matrix LG is positive semidefinite If the graph G is connected, then the matrix L is positive definite Thus its inverse L exists Let the i, j-entry of L be denoted by t ij Then, x t =L b t, implying x k = l Γ n t kl 3 Proof of Theorem As already explained, it is sufficient to demonstrate the validity of Theorem for i = n j = n By 2 3 we have under the extreme value condition f x,x 2,,x n =d n k Γ n t kl 4 l Γ n Let L k,l be the submatrix obtained by removing from L n the k-th row the l-th column Then, t kl = k+l detl k,l detl Thus by 4 f x,x 2,,x n 5 = d n k Γ n l Γ n k+l detl k,l detl By exping detln with respect to its last column we get detln=d n detl k+n detlnk,n, k Γ n where the submatrix Lnk,n is obtained by removing the k-th row the n -th column of Ln Further, exping detlnk,n with respect its last row, det Lnk, n = This yields d n detl = detln, l Γ n k Γ n l Γ n which substituted back into 5 becomes l+n detl k,l k+l detl k,l f x,x 2,,x n = detln detl 6 Theorem follows when 6 is substituted back into 7

4 R B Bapat et al A Simple Method for Computing Resistance Distance Applications The most obvious probably the most useful application of formula 6 is in a simple procedure for computing resistance distances If, 2,, n, 2,, are the eigenvalues of the submatrices Li Li, j, respectively, then n det Li= r= detli, j= r, 7 r= r, 8 r ij is readily obtained As far as algorithm complexity is concerned, the usage of 7 8 is not the most efficient way to compute resistance distances However, the method is extremely simple for writing a computer program provided, of course, that a software for matrix diagonalization is available Corollary For any connected n-vertex graph, n 2, the Kirchhoff index, is expressed in terms of Laplacian eigenvalues as 5 Proof Suppose that the Laplacian characteristic polynomial of G, defined as ψg,λ =detλ I n LG, where I n is the unit matrix of order n, is written as ψg,λ = n k=0 k c k λ n k Then all the coefficients c k are non-negative integers, inparticular [24, 26] c n = 0, c n = ntg, c f 4, c = n j= j n i= i n = ntg i= i The Kirchhoff index is defined via Then by Theorem, detli, j Kf = r ij = = c i< j i< j tg tg n = n i i= Formula 5 was reported earlier for details further references see [7, 33], but was deduced by a completely different way of reasoning Since r ij is a distance function [25], we obtain by Theorem : Corollary 2 If i, j,k are distinct to each other, then detli,k detli, j+detl j,k If i, j are distinct, then by the well-known Hadamard-Fisher inequality detli d i detli, j Thus from Theorem immediately follows Corollary 3 { r ij max, } d i d j In the below Theorem 2 we deduce a better lower, also an upper bound for the resistance distance From 9 we get bl b t = bx t = Then by 2, k= b k x k = k Γ n x k f x,x 2,,x n =d n bl b t 9 Evidently, if the vertices v n v n are adjacent then d n = bb t +, otherwise d n = bb t Let I be the unit matrix of order n 2 Then by 9 we obtain f x,x 2,,x n =b[i L ]b t 20 if the vertices v n v n are not adjacent Otherwise [ ] dn f x,x 2,,x n =b d n I L b t 2 As before, the eigenvalues of the matrix Li, j are denoted by 2 > 0 Then, of course, / i, i =,2,,n 2, are the eigenvalues of L Then from 20 2 by taking into account 7 follows: Theorem 2 Let i j If the vertices v i v j are not adjacent, then d i d i r ij

5 498 R B Bapat et al A Simple Method for Computing Resistance Distance d i r ij d i with the upper bound for r ij applicable only if d i > 0,ie, > Otherwise d i + d i r ij d i + r ij d i + + [] R E Merrifield H E Simmons, Theor Chim Acta 55, [2] R E Merrifield H E Simmons, Topological Methods in Chemistry, Wiley, New York 989 [3] I Gutman O E Polansky, Mathematical Concepts in Organic Chemistry, Springer-Verlag, Berlin 986, pp 2 6 [4] O E Polansky, Z Naturforsch 4a, [5] M Zer, Z Naturforsch 45a, [6] B Lučić, I Lukovits, S Nikolić, N Trinajstić, J Chem Inf Comput Sci 4, [7] A A Dobrynin, R Entringer, I Gutman, Acta Appl Math 66, [8] A A Dobrynin, I Gutman, S Klavžar, P Žigert, Acta Appl Math 72, [9] D H Rouvray W Tatong, Z Naturforsch 4a, [0] I Gutman T Körtvélyesi, Z Naturforsch 50a, [] I Gutman I G Zenkevich, Z Naturforsch 57a, [2] D J Klein M Rić, J Math Chem 2, [3] D Bonchev, A T Balaban, X Liu, D J Klein, Int J Quantum Chem 50, 994 [4] H Y Zhu, D J Klein, I Lukovits, J Chem Inf Comput Sci 36, [5] I Gutman B Mohar, J Chem Inf Comput Sci 36, [6] D J Klein, MATCH Commun Math Comput Chem 35, with the upper bound for r ij applicable only if d i + > 0,ie, > /d i Acknowledgements This research was supported by the Natural Science Foundation of China Fujian Province, by the Ministry of Sciences, Technologies Development of Serbia, within the Project no 389 [7] I Lukovits, S Nikolić, N Trinajstić, Int J Quantum Chem 7, [8] D J Klein O Ivanciuc, J Math Chem 30, [9] J L Palacios, Int J Quantum Chem 8, [20] D J Klein, Croat Chem Acta 75, [2] D Babić, D J Klein, I Lukovits, S Nikolić, N Trinajstić, Int J Quantum Chem 90, [22] W J Xiao I Gutman, MATCH Commun Math Comput Chem 49, [23] W J Xiao I Gutman, Theor Chim Acta, in press [24] B Bollobás, Modern Graph Theory, Springer-Verlag, New York 998, Chapter 9 [25] R B Bapat, Math Student 68, [26] D M Cvetković, M Doob, H Sachs, Spectra of Graphs-Theory Application, Academic Press, New York 980 [27] R B Bapat, Math Student 65, [28] R B Bapat D M Kulkarni, in: D V Huynh, S K Jain, S R Lopez-Permouth Eds, Algebra Its Applications, Amer Math Soc, Providence 2000, pp [29] I Gutman R B Mallion, Z Naturforsch 48a, [30] R Grone, R Merris, V S Sunder, SIAM J Matrix Anal Appl, [3] R Merris, Lin Algebra Appl 97, [32] R Merris I Gutman, Z Naturforsch 55a, [33] I Gutman, D Vidović, B Furtula, Indian J Chem 42A,

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