Linear Algebra and its Applications

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1 Linear Algebra and its Applications Contents lists available at ScienceDirect Linear Algebra and its Applications journal homepage: wwwelseviercom/locate/laa On the characteristic and Laplacian polynomials of trees Abbas Heydari a, Bijan Taeri b, a b Department of Mathematics, Islamic Azad University, Arak , Iran Department of Mathematical Sciences, Isfahan University of Technology, Isfahan , Iran A R T I C L E I N F O A B S T R A C T Article history: Received 27 November 2007 Accepted 8 September 2009 Available online 13 October 2009 Submitted by S Kirkland Keywords: Characteristic polynomial Tree Adjacency matrix Laplacian matrix We find the characteristic polynomials of adjacency and Laplacian matrices of arbitrary unweighted rooted trees in term of vertex degrees, using the concept of the rooted product of graphs Our result generalizes a result of Rojo and Soto [O Rojo, R Soto, The spectra of the adjacency matrix and Laplacian matrix for some balanced trees, Linear Algebra Appl ] on a special class of rooted unweighted trees, namely the trees such that their vertices in the same level have equal degrees 2009 Elsevier Inc All rights reserved 1 Introduction and results Computing characteristic polynomials of adjacency and Laplacian matrix of graphs is one of the essential topics in the theory of graphs Trees, which are acyclic connected graphs, are widely used in various fields of sciences A well known formula for the characteristic polynomial of a tree with n vertices is i c i λ n i, where the odd coefficients are zero, and the even coefficients are given by the rule that 1 r c 2r is the number of ways of choosing r disjoint edges in the tree see [1, p 49] For a rooted tree of three levels there is an exact formula for the characteristic polynomial of its adjacency matrix in terms of vertex degrees see for example [10] Let G be a simple graph Throughout the paper AG, DG, and LG = DG AG, denote the adjacency matrix, the diagonal matrix of vertex degrees, and the Laplacian matrix of G, respectively The characteristic polynomials of AG and LG are denoted by P G λ and Q G λ, respectively Finally we denote the spectrum of a matrix M by σm Let x 1 be a vertex of degree 1 in the graph G and let x 2 be the vertex adjacent to x 1 LetG 1 be the induced subgraph obtained from G by deleting the vertex x 1 and G 2 be the induced subgraph obtained from G by deleting the vertices x 1 and x 2 Then see [2, p 59] Corresponding author addresses: a-heydari@mathiutacir A Heydari, btaeri@cciutacir B Taeri /$ - see front matter 2009 Elsevier Inc All rights reserved doi:101016/jlaa

2 662 A Heydari, B Taeri / Linear Algebra and its Applications P G λ = λp G1 λ P G2 λ By iterating the above formula, the characteristic polynomial of a tree can easily be determined Recently computing characteristic polynomials of adjacency and Laplacian matrices of some classes of trees has been the object of many papers see for example [5 9] Rojo and Soto in [6] obtained characteristic polynomials of adjacency and Laplacian matrices of a special kind of rooted unweighted trees, namely the trees such that their vertices in the same level have equal degrees In this paper we use the concept of rooted product of graphs and find a recursive formula for characteristic polynomials of adjacency and Laplacian matrices of arbitrary rooted unweighted trees, in terms of vertex degrees Let us recall rooted product of graphs Suppose that G ={G 1, G 2,, G k } is a sequence of k rooted graphs and H is a labelled graph on k vertices The rooted product of H by G, which is denoted by HG, is obtained by identifying the root vertex of G i with the ith vertex of H see [4] In [3] the characteristic polynomials of AHG and LHG are computed, in terms of the characteristic polynomials of the graphs H and G i, i = 1, 2,, k One can represent a rooted tree in terms of a suitable rooted product In fact let T bearootedtreeof k + 1 levels and G 1, G 2,, G n be rooted trees of k levels that are obtained by deletion the root vertex of T Then T = S n+1 S 1, G 1, G 2,, G n is the rooted product of S n+1 by {S 1, G 1, G 2,, G n }, where S m is the star on m vertices see Fig 1 In order to state the main results of the paper we need some notation Let T bearootedtree of k levels Suppose that n k j+1, where j = 1, 2,, k, is the number of vertices on level j and d k j+1,i, where i = 1, 2,, n k j+1, is the degree of the ith vertex of level j Ife j,i denotes the number of neighbors on level k j + 2oftheith vertex from the level k j + 1ofT, where 2 j k, then { dj,i if j = k, e j,i = d j,i 1 if j /= k For j = 1, 2, 3,, k, put m j,0 = 0 and for 1 r n k j+1 put m j,r = e k j+1,1 + +e k j+1,r Inthe following theorem, we find a recursive formula for the characteristic polynomial of the adjacency matrix of a rooted tree of k levels Fig 1 S 4 S 1, G 1, G 2, G 3, where S n+1 is the star on n vertices and G 1, G 2, G 3 are arbitrary rooted trees

3 A Heydari, B Taeri / Linear Algebra and its Applications Theorem 1 LetTbearootedtreeofklevels Suppose that P 0,i = 1, where i = 1, 2,, n 1, and P 1,i = λ, where i = 1, 2,, n 2 If e j+1,r = 0 we put P j+1,r = λ, where j = 1, 2,, k 1 and 1 r n j In other cases we put m j+1,r P j+1,r = P j,i λ +m j+1,r 1 Then P T λ = P k,1 m j+1,r mj,i s=1+m j,i 1 P j 1,s +m j+1,r 1 P j,i Using a similar argument as in the proof of Theorem 1, we can derive a recursive formula for Laplacian polynomial of a rooted tree Theorem 2 LetTbearootedtreeofklevels Suppose that Q 0,i = 1, where i = 1, 2,, n 1 and Q 1,i = λ 1, where i = 1, 2,, n 2 Ife j+1,r = 0 we put Q j+1,r = λ 1, where j = 1, 2,, k 1 and 1 r n j In other cases we put m j+1,r Q j+1,r = Q j,i λ d j+1,r +m j+1,r 1 Then Q T λ = Q k,1 m j+1,r mj,i s=1+m j,i 1 Q j 1,s +m j+1,r 1 Q j,i 2 Proofs In this section we prove Theorem 1 Also we state some corollaries and examples of Theorems 1 and 2 Let G ={G 1, G 2,, G k } be a sequence of k rooted graphs of order n 1, n 2,, n k respectively, and H be a labelled graph on k vertices We shall use a suitable labelling of the vertices of HG 1, G 2,, G k as follows The root vertex of G 1 has label 1 and the vertices of G 1 have consecutive labels from 1 to n 1 The root vertex of G 2 has label n and the vertices of G 2 have consecutive labels from n to n 1 + n 2,, and finally the root vertex of G k has label n 1 + n 2 + +n k and the vertices of G k have consecutive labels from n 1 + n 2 + +n k 1 + 1ton 1 + n 2 + +n k Now let M 1,1 denote the matrix obtained by deletion of the first column and row of a matrix M, and put P Gi λ = detλi AG i 1,1, Q Gi λ = detλi LG i 1,1 IfG i is the graph of order 1 put P Gi λ = Q Gi λ = 1 We need the following theorem which is proved in [3, Theorem 1] the first part of Theorem A is proved by Godsil and McKay in [4]: Theorem A Let H and G 1, G 2,, G k, j = 1, 2,, k, be simple graphs If K = HG 1, G 2,, G k then P K λ = 1 k detm, where P Gi λ if i = j, M ij = P Gi λ if i /= j and AH ij = 1, 0 if i /= j and AH ij = 0, 1 i, j k, and Q K λ = detn, where Q Gi λ if i = j, N ij = Q Gi λ 0 if i /= j and AH ij = 1, if i /= j and AH ij = 0, 1 i, j k We also need the following lemma, which can be proved by an easy induction on n

4 664 A Heydari, B Taeri / Linear Algebra and its Applications Lemma 1 For i = 1, 2,, n, let x i be an arbitrary variable Then x x x 3 0 n n n = x i x j i=2 j=2,j /=i x n n n Now we are ready to prove a main result of the paper Proof of Theorem 1 We prove the assertion by induction on k Firstletk = 2 Then T isastaron n vertices and n 1 n 1 1 P T λ = P 2,1 = P 1,i λ = λ n1 1 λ 2 n 1 P 1 1,i Thus the assertion is true for k = 2 Suppose that the result is true for all positive integers which are smaller than k Suppose that T isarootedtreeofk levels Let G 1, G 2,, G ek be rooted trees with root vertex degrees e k 1,1, e k 1,2, e k 1,ek that are obtained by deletion the root vertex of T Then T = S ek +1S 1, G 1, G 2,, G ek Since AS ek +1 = using the notation of Theorem A, we have M 11 = P S1 λ = λ, M ii = P Gi 1 λ, 2 i e k + 1, M 1j = P S1 λ = 1, 2 j e k + 1, M i1 = P Gi 1 λ, 2 i e k + 1 Therefore by Theorem A and Lemma 1 we have λ P G1 λ P G1 λ 0 0 P T λ = 1 e k+1 P G2 λ 0 P G2 λ 0 P Gek λ 0 0 P Gek λ λ P G 1 λ 0 0 P G1 λ = 1 e k+1 PG1 λp G2 λ P Gek λ 1 0 P G 2 λ 0 P G2 λ P Ge k λ P Gek λ

5 A Heydari, B Taeri / Linear Algebra and its Applications = P G1 λp G2 λ P Gek λ λ = e k P Gi λ λ e k e k ek P Gi λ P Gi λ e k j=1,j /=i P Gj λ P Gj λ P Gi λ 1 P Gi λ By the induction hypothesis P Gr λ = P k 1,r, for 1 r e k Since P Gr λ is the characteristic polynomial of the graph which obtained by deleting the root vertex of G r, for 1 r e k,wehave m k 1,r P Gr λ = P k 2,s s=1+m k 1,r 1 In the special case if P Gr λ = λ, using the notation of Theorem A, wehavep Gr λ = 1 Since in the first level of T there is one vertex, we have n k = 1, m k,1 = e k Thus by replacing the last results in 1 we obtain that m k,1 m k,1 mk 1,i s=1+m P T λ = P k 1,i λ k 1,i 1 P k 2,s P +m k,0 +m k,0 k 1,i Therefore P T λ = P k,1 and the Theorem is proved Taking k = 3 in Theorem 1, we can obtain a well known formula, computed in [10], on the characteristic polynomial of the adjacency matrix of a rooted tree of 3 levels A rooted tree of 3 levels is shown in Fig 2 Corollary 1 LetTbearootedtreeof3levels and n = e 3,1 Then P T λ = λ n n n e 2,i n 1 λ 2 1 e 2,i 1 λ 2 e 2,i Proof Put k = 3 By Theorem 1, P T λ = P 3,1 Fori = 1, 2,, n 2, P 2,i = λ e 2,i 1 λ 2 e 2,i is the characteristic polynomial of the star graph of order e 2,i + 1 and for j = 1, 2,, n 1, P 1,j = λ Thus if n = e 3,1 denotes the degree of root vertex of T, then Fig 2 A rooted tree of 3 levels

6 666 A Heydari, B Taeri / Linear Algebra and its Applications m2,i n n s=1+m P T λ = P 2,i λ 2,i 1 P 1,s P 2,i n n λ = λ e2,i 1 λ 2 e 2,i λ e 2,i λ e 2,i 1 λ 2 e 2,i = λ n n n e 2,i n 1 λ 2 1 e 2,i 1 λ 2 e 2,i This completes the proof Now suppose T isarootedtreeofklevels and the vertices of T in the same level have equal degree In [8] Rojo and Robbiano, called such a tree a generalized Bethe tree They denoted the class of generalized Bethe trees of k levels by B k Rojo and Soto [6] computed the characteristic polynomial of adjacency and Laplacian matrices of graphs in B k As a corollary of Theorem 1 we can compute P βk λ, where β k is a generalized Bethe tree of k levels If d k j+1 denotes the vertex degree of vertices on level j in β k, then put { dj if j = k, e j = d j 1 if j /= k Since the vertices on the jth level in β k have equal degree, P j,1 = P j,i, for all i = 2,, n k j+1 IfP j λ denotes P j,i, then by Theorem 1 P j λ = P e j 1 λ j 1 λp j 1 λ e j P e j 1 j 2 λ 2 Corollary 2 Let P 0 λ = 1, P 1 λ = λ and P j λ = λp j 1 λ e j P j 2 λ for all j = 2, 3,, k Then P βk λ = P k λ k 1 j=1 Pn j n j+1 j λ Proof The proof is done by using 2 and an easy induction on k Now we state some corollaries of Theorem 2 First we find an exact formula for the characteristic polynomial of the Laplacian matrix of a rooted tree of 3 levels Corollary 3 LetTbearootedtreeof3levels and n = e 3,1 The characteristic polynomial of the Laplacian matrix of T is given by Q T λ = λ 1 n n n λ e 2,i n λ e 2,i λ + 1 λ n λ e 2,i λ + 1 Proof By Theorem 2, we have to compute Q 3,1 Forj = 1, 2,, n 1,wehaveQ 1,j = λ 1 and for r = 1, 2,, n 2 we have m 2,r m 2,r 1 Q 2,r = λ d 2,r = Q 1,i +m 2,r 1 e 2,r Q +m 2,r 1 1,i e 2,r 1 λ 1 λ e 2,r 1 λ 1 = λ 1 e 2,r 1 λ e 2,i λ + 1

7 A Heydari, B Taeri / Linear Algebra and its Applications Now let n = e 3,1 denotes the degree of root vertex of T Thus m2,i n n s=1+m Q T λ = Q 2,i λ n 2,i 1 Q 1,s Q 2,i n = λ 1 e 2,i 1 λ e 2,i + 1 n λ 1 λ e 2,i n λ 1 e 2,i 1 λ e 2,i + 1 = λ 1 n n n λ e 2,i n λ e 2,i λ + 1 λ n λ e 2,i λ + 1 So the proof is complete Suppose that β k is a generalized Bethe tree of k levels Since the vertices on the jth level in β k have equal degree, P j,1 = P ji, for all i = 1, 2,, n k j+1 IfP j λ denotes P j,i, then by Theorem 2 Q j λ = Q e j 1 j 1 λ λ d j Q j 1 e j Q e j 1 j 2 λ 3 Using 3 we can compute the characteristic polynomial of the Laplacian matrix of β k Corollary 4 Let Q 0 λ = 1, Q 1 λ = λ 1 and Q j λ = λ d j Q j 1 λ e j Q j 2 λ for all j = 2, 3,, k Then Q βk λ = Q k λ k 1 j=1 Q n j n j+1 j λ Proof The proof can be done by using 3 and an easy induction on k Fig 3 A rooted tree of 4 levels

8 668 A Heydari, B Taeri / Linear Algebra and its Applications Example 1 Let T be a rooted tree of 4 levels as shown in Fig 3 InT we have n 1 = n 2 = 6, n 3 = 4, n 4 = 1 and e 2,1 = e 2,2 = e 2,3 = 0, e 2,4 = 1, e 2,5 = 3, e 2,6 = 2 The values of e i,j and the labelling of T are shown in Fig 3 To compute the characteristic polynomial of the adjacency matrix of T, using Theorem 1, we must calculate P 4,1 = P T λ For vertices on the third level of T we have P 2,1 = P 2,2 = P 2,3 = λ By Theorem 1wehave P 2,4 = P 1,1 λ 1 = λ 2 1, P 1,1 P 2,5 = P 1,2 P 1,3 P 1,4 P 2,6 = P 1,5 P 1,6 λ 1 P 1,5 1 λ P 1,2 P 1,3 P 1,4 P 1,6 = λλ 2 2 For vertices on the second level of T we have P 3,1 = λ and P 3,2 = P 2,1 P 2,2 λ 1 1 = λλ 2 2, P 2,1 P 2,2 = λ 2 λ 2 3, P 3,3 = P 2,3 P 2,4 λ 1 P 1,1 = λ 4 3λ 2 + 1, P 2,3 P 2,4 P 3,4 = P 2,5 P 2,6 λ P 1,2P 1,3 P 1,4 P 1,5P 1,6 = λ 8 7λ λ 4 P 2,5 P 2,6 Therefore P T λ can be computed as follows: P 4,1 = P 3,1 P 3,2 P 3,3 P 3,4 λ 1 P 2,1P 2,2 P 2,3P 2,4 P 2,5P 2,6 P 3,1 P 3,2 P 3,3 P 3,4 = λ 17 16λ λ λ λ 9 198λ λ 5 To compute the characteristic polynomial of the Laplacian matrix of T, by Theorem 2, we have to calculate Q 4,1 = Q T λ For vertices on the third level of T we have Q 2,1 = Q 2,2 = Q 2,3 = λ 1 By Theorem 2 we have Q 2,4 = Q 1,1 λ 1 1 = λ 2 2λ, Q 1,1 Q 2,5 = Q 1,2 Q 1,3 Q 1,4 λ = λλ 4λ 1 2, Q 1,2 Q 1,3 Q 1,4 Q 2,6 = Q 1,5 Q 1,6 λ = λλ 1λ 3 Q 1,5 Q 1,6 For vertices on the second level of T we have Q 3,1 = λ 1, Q 3,2 = Q 2,1 Q 2,2 λ 2 1 Q 2,1 1 Q 2,2 = λλ 1λ 3,

9 A Heydari, B Taeri / Linear Algebra and its Applications Q 3,3 = Q 2,3 Q 2,4 λ 2 1 Q 1,1 = λλ 2λ 2 4λ + 2, Q 2,3 Q 2,4 Q 3,4 = Q 2,5 Q 2,6 λ 2 Q 1,2Q 1,3 Q 1,4 Q 1,5Q 1,6 Q 2,5 Q 2,6 = λλ 1 3 λ 4λ 3 7λ λ 2 Therefore Q T λ can be computed as follows: Q 4,1 = λ 17 32λ λ λ λ λ λ λ λ λ λ λ λ λ λ 3 450λ λ Acknowledgements The authors are greatly indebted to the referee whose valuable remarks and suggestions helped to revise the paper The authors also thank Professor Kirkland for his kind cooperations during the reviewing process of the article The second author was partially supported by the Center of Excellence of Mathematics of Isfahan University of Technology CEAMA References [1] N Biggs, Algebraic Graph Theory, Cambidge University, 1993 [2] D Cvetkovic, M Doob, H Sachs, Spectra of Graphs Theory and Application, third ed, Johann Ambrosius Barth Verlag, Heidelberg, Leipzig, 1995 [3] A Heydari, B Taeri, On the characteristic polynomial of a special class of graphs and spectra of balanced trees, Linear Algebra Appl [4] CD Godsil, BD McKay, A new graph product and its spectrum, Bull Austral Math Soc [5] O Rojo, The spectrum of the Laplacian matrix of a balanced binary tree, Linear Algebra Appl [6] O Rojo, R Soto, The spectra of the adjacency matrix and Laplacian matrix for some balanced trees, Linear Algebra Appl [7] O Rojo, On the spectra of certain rooted trees, Linear Algebra Appl [8] O Rojo, M Robbiano, On the spectra of some weighted rooted trees and applications, Linear Algebra Appl [9] O Rojo, The spectra of a graph obtained from copies of a generalized Bethe tree, Linear Algebra Appl [10] L Wang, X Li, S Zhang, Families of integral trees with diameters 4, 6, and 8, Discrete Appl Math