Copies of a rooted weighted graph attached to an arbitrary weighted graph and applications

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1 Electronic Journal of Linear Algebra Volume 26 Volume Article Copies of a rooted weighted graph attached to an arbitrary weighted graph and applications Domingos M. Cardoso dcardoso@ua.pt Enide A. Martins Maria Robbiano Oscar Rojo Follow this and additional works at: Recommended Citation Cardoso, Domingos M.; Martins, Enide A.; Robbiano, Maria; and Rojo, Oscar. 2013, "Copies of a rooted weighted graph attached to an arbitrary weighted graph and applications", Electronic Journal of Linear Algebra, Volume 26. DOI: This Article is brought to you for free and open access by Wyoming Scholars Repository. It has been accepted for inclusion in Electronic Journal of Linear Algebra by an authorized editor of Wyoming Scholars Repository. For more information, please contact scholcom@uwyo.edu.

2 COPIES OF A ROOTED WEIGHTED GRAPH ATTACHED TO AN ARBITRARY WEIGHTED GRAPH AND APPLICATIONS DOMINGOS M. CARDOSO, ENIDE A. MARTINS, MARIA ROBBIANO, AND OSCAR ROJO Abstract. The spectrum of the Laplacian, signless Laplacian and adjacency matrices of the family of the weighted graphs R{H}, obtained from a connected weighted graph R on r vertices and r copies of a modified Bethe tree H by identifying the root of the i-th copy of H with the i-th vertex of R, is determined. Key words. Weighted graph, Generalized Bethe tree, Laplacian matrix, Signless Laplacian matrix, Adjacency matrix, Randić matrix. AMS subject classifications. 05C50, 15A Introduction. Let G = VG, EG be a simple undirected graph with vertex set VG = {1,...,n} and edge set EG. We assume that each edge e EG has a positive weight we. The adjacency matrix AG = a i,j of G is the n n matrix in which a i,j = we if there is an edgeejoining i and j and a i,j = 0 otherwise. Let DG be the diagonal matrix in which the diagonal entry d i,i = ewe where the sum is over all the edges e incident to the vertex i. The Laplacian matrix and the signless Laplacian matrix of G are LG = DG AG and QG = DG+AG, respectively. The matrices LG, QG and AG are real and symmetric. From Geršgorin s theorem, it follows that the eigenvalues of LG and QG are nonnegative real numbers. Since the rows of LG sum to 0, 0,e is an eigenpair for LG, where e is the all ones vector. Fiedler [9] proved that G is a connected graph if and only Received by the editors on November 1, Accepted for publication on September 22, Handling Editor: Xingzhi Zhan. CIDMA - Centro de Investigação e Desenvolvimento em Matemática e Aplicações, Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal dcardoso@ua.pt, enide@ua.pt. Research supported in part by FEDER funds through COMPETE Operational Programme Factors of Competitiveness Programa Operacional Factores de Competitividade and by Portuguese funds through the Center for Research and Development in Mathematics and Applications and the Portuguese Foundation for Science and Technology FCT Fundação para a Ciência e a Tecnologia, within project PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP FEDER Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile mrobbiano@ucn.cl. Research partially supported by Fondecyt - IC Project , Chile. Departamento de Matemáticas, Universidad Católica del Norte, Antofagasta, Chile orojo@ucn.cl. Research supported by Project Fondecyt Regular and Project Fondecyt Regular , Chile. This author thanks the hospitality of the Departamento de Matemática, Universidade de Aveiro, Aveiro, Portugal, in which this research was finished. All the authors are members of the Project PTDC/MAT/112276/

3 Copies of a Rooted Weighted Graph Attached to an Arbitrary Weighted Graph 707 if the second smallest eigenvalue of LG is positive. This eigenvalue is called the algebraic connectivity of G. The signless Laplacian matrix has recently attracted the attention of several researchers and some papers on this matrix are [2, 4, 5, 6, 7]. In this paper, M G is one of the matrices LG, QG or AG. If we = 1 for all e EG then G is an unweighted graph. Let R be a connected weighted graph on r vertices. Let v 1,v 2,...,v r be the vertices of R. As usual, v i v j means that v i and v j are adjacent. Let ε i,j = ε j,i be the weight of the edge v i v j if v i v j, and let ε i,j = ε j,i = 0 otherwise. Moreover, for i = 1,2,...,r, let ε i = v j v i ε i,j. Let R{H} be the graph obtained from R and r copies of a rooted weighted graph H by identifying the root of i copy of H with v i. Example 1.1. If R is the graph depicted in Figure 1.1 and H is the graph Fig The cycle C 4. depicted in Figure 1.2 then R{H} is the graph depicted in Figure 1.3. Root R r r Fig A modified Bethe tree, H, with four levels. We recallthat forarootedgraphthe level ofavertexis one morethan its distance from the root vertex. Let B be a weighted generalized Bethe tree of k > 1 levels, that is, B isarootedtreeinwhichverticesatthe samelevelhavethesamedegreeandedges connecting vertices at consecutive levels have the same weight. Consider a nonempty subset {1,2,...,k 1} and a family of graphs F = {G j : j }. For j, we assume that the edges of G j have weight u j. Let BF be the graph obtained from B and the graphs in F identifying each set of children of B at level k j +1 with the vertices of G j.

4 R r r R r r Electronic Journal of Linear Algebra ISSN D.M. Cardoso, E.A. Martins, M. Robbiano, and O. Rojo R r r R r r Fig The graph R{H}. Example 1.2. Let BF be the graph depicted in Figure 1.2. In this graph, B is a generalized Bethe tree of k = 4 levels, = {1,3}, F={G 1,G 3 }, where G 1 is a star of 4 vertices and G 3 is a path of 2 vertices. In this paper, we derive a general result on the spectrum of M R{H}. Using this result, we characterize the eigenvalues of the Laplacian matrix, including their multiplicities, of the graph R{BF}; and also of the signless Laplacian and adjacency matrices whenever the subgraphs in F are regular. They are the eigenvalues of symmetric tridiagonal matrices of order j, 1 j k. In particular, the Randić eigenvalues are characterized. Denote by σc the multiset of eigenvalues of a square matrix C. 2. A result on the spectrum of M R{H}. Let E be the matrix of order n n with 1 in the n,n entry and zeros elsewhere. For i = 1,2,...,r, let dv i be the degree of v i as a vertex of R and let n be the order of H. Then R{H} has rn vertices. We label the vertices of R{H} as follows: for i = 1,2,...,r, using the labels i 1n+1,i 1n+2,...,in, we label the vertices of the i th copy of H from the bottom to the vertex v i and, at each level, from the left to the right. With this labeling, M R{H} is equal to M H+aε 1 E sε 1,2 E sε 1,r E. sε 1,2 E..... sε2,r E ,.... M H+aεi,r 1 E sε r 1,r E sε 1,r E sε 2,r E sε r 1,r E M H+aε r E

5 Copies of a Rooted Weighted Graph Attached to an Arbitrary Weighted Graph 709 where 2.2 and 2.3 s = a = { 1 if M is the Laplacian matrix, 1 if M is the signless Laplacian or adjacency matrix { 0 if M is the adjacency matrix, 1 if M is the Laplacian or signless Laplacian matrix. In this paper, the identity matrix of appropriate order is denoted by I and I m denotes the identity matrix of order m. Furthermore, A denotes the determinant of the matrix A and A T is the transpose of A. We recall that the Kronecker product [12] of two matrices A = a i,j and B = b i,j of sizes m m and n n, respectively, is defined as the mn mn matrix A B = a i,j B. Then, in particular, I n I m = I nm. Some basic properties of the Kronecker product are A B T = A T B T and A BC D = AC BD for matrices of appropriate sizes. Moreover, if A and B are invertible matrices then A B 1 = A 1 B 1. Theorem 2.1. Let ρ 1 R,ρ 2 R,...,ρ r R be the eigenvalues of M R. Then 2.4 σm R{H} = r s=1σm H+ρ s RE. Proof. From 2.1, it follows M R{H} = I r M H+M R E. Let V = [ v 1 v 2 v r 1 v r ] be an orthogonal matrix whose columns v 1,v 2,...,v r are eigenvectors corresponding to the eigenvalues ρ 1 R,ρ 2 R,...,ρ r R, respectively. Then V I n M R{H} V T I n = V In I r M H+M R E V T I n = I r M H+ VM RV T E. We have VM RV T E = ρ 1 R ρ 2 R ρ r R E

6 710 D.M. Cardoso, E.A. Martins, M. Robbiano, and O. Rojo ρ 1 RE ρ 2 RE. =..... ρ r RE. Therefore, V I n M R{H} V T I n M H+ρ 1 RE M H+ρ 2 RE = M H+ρ r RE. Since M R{H} and V I n M R{H} V T I n are similar matrices, we conclude Application to a modified generalized Bethe tree with subgraphs at some levels. Let B be a weighted generalized Bethe tree of k levels k > 1. For 1 j k, n j and d j are the number and the degree of the vertices of B at the level k j + 1, respectively. Thus, d k is the degree of the root vertex assumed greater than 1, n k = 1, d 1 = 1 and n 1 is the number of pendant vertices. For 1 j k 1, let w j be the weight of the edges connecting the vertices of B at the level k j +1 with the vertices at the level k j. We consider δ j = w j if j = 1, d j 1w j 1 +w j if 2 j k 1, d k w k 1 if j = k. We observe that δ j is the sum of the weights of the edges of B incident with a vertex at the level k j+1 and if w 1 = w 2 = = w k 1 = 1 then δ j = d j for j = 1,2,...,k. Let m j = nj n j+1 for j = 1,2,...,k 1. Then m j = d j j k 2, d k = n k 1 = m k 1. Note that m j is the cardinality of each set of children at the level k j +1. Let M j = M G j. From now on, we assume that µ 1 M j,...,µ mj M j are the eigenvalues of M j and e mj is an eigenvector for µ mj M j, that is, 3.1 M j e mj = µ mj M j e mj.

7 Copies of a Rooted Weighted Graph Attached to an Arbitrary Weighted Graph 711 We observe that 3.1 holds when M G j = LG j and when M G j = QG j or M G j = AG j if G j is a regular graph. Assuming 3.1, in [11], we characterizethe eigenvalues ofthe matrix M BF = In 2 S 1 sin 2 w 1 em 1 sin 2 w 1 e T m Ink 1 S k 2 sin w k 1 k 2 em k 2 sin w k 1 k 2 em k 2 S k 1 sw k 1 em k 1 sw k 1 em k 1 aδ k in which, for j = 1,2,...,k 1, 3.2 S j = { aδj I mj +M G j if j, aδ j I mj if j /, with s and a as in 2.2 and 2.3. The results in [11] generalize several previous contributions see [3, 8, 10]. Definition 3.1. For j = 1,...,k, let X j be the j j leading principal submatrix of the k k symmetric tridiagonal matrix X k = aδ 1 +µ m1 M 1 w 1 m1. w 1 m1 aδ 2 +µ m2 M wk 2 mk 2 w k 2 mk 2 aδ k 1 +µ mk 1 M k 1 w k 1 mk 1 w k 1 mk 1 aδ k Definition 3.2. For j = 1,2,...,k 1 and i = 1,...,m j 1, let X j,i = aδ 1 +µ m1 M 1 w 1 m1. w 1 m wj 2 mj 2 w j 2 mj 2 aδ j 1 +µ mj 1 M j 1 w j 1 mj 1 w j 1 mj 1 aδ j +µ i M j Finally, let Ω = {j : 1 j k 1, n j > n j+1 }. We are ready to state the main result published in [11]. Theorem 3.3. [11] σm BF = σx k j Ω σx j nj nj+1 j mj 1 i=1 σx j,i nj+1,

8 712 D.M. Cardoso, E.A. Martins, M. Robbiano, and O. Rojo where σx j nj nj+1 and σx j,i nj+1 mean that each eigenvalue in σx j and in σx j,i must be considered with multiplicity n j n j+1 and n j+1, respectively. Furthermore, the multiplicities of equal eigenvalues obtained in different matrices if any, must be added. An equivalent version of Theorem 3.3 is: Theorem 3.4. λi M BF = D k λ j Ω D j λ nj nj+1 j m j 1 i=1 D j,i λ nj+1, where, for j = 1,2,...,k and i = 1,2,...,m j 1, D j λ and D j,i λ are the characteristic polynomials of the matrices X j and X j,i, respectively. Let Ã be the submatrix obtained from a square matrix A by deleting its last row and its last column. Moreover, from the proofs of Lemma 2.2, Theorem 2.5 and Lemma 2.7 in [11], we obtain: Lemma 3.5. λi M BF = D k 1 λ j Ω D j λ nj nj+1 j m j 1 i=1 D j,i λ nj+1. Definition 3.6. For s = 1,2,...,r, let Y s = aδ 1 +µ m1 M 1 w 1 m1 w 1 m1 aδ 2 +µ m2 M wk 2 mk 2 w k 2 mk 2 aδ k 1 +µ mk 1 M k 1 w k 1 mk 1 w k 1 mk 1 aδ k +ρ sr. We are ready to apply Theorem 2.1 to H = BF. Theorem 3.7. If G = R{BF}, then σm G = j Ω σx j rn j n j+1 j m j 1 i=1 σx j,i rn j+1 r s=1σy s, where σx j rnj nj+1 and σx j,i rnj+1 mean that each eigenvalue in σx j and in σx j,i must be considered with multiplicity rn j n j+1 and rn j+1, respectively. Furthermore, the multiplicities of equal eigenvalues obtained in different matrices if any, must be added.

9 Copies of a Rooted Weighted Graph Attached to an Arbitrary Weighted Graph 713 Proof. For H = BF, from Theorem 2.1, we have 3.3 σm R{BF} = r s=1σm BF+ρ s RE. Let 1 s r. By linearity on the last column, we have λi M BF ρ s RE = λi M BF ρ s R λi M BF. Using Theorem 3.4 and Lemma 3.5, λi M BF ρ s RE has the form D k λ ρ s RD k 1 λ j Ω Now, by linearity on the last column, we have D j λ nj nj+1 j m j 1 i=1 D j,i λ nj+1. λi Y s = λi X k ρ s R λi X k 1 = D k λ ρ s RD k 1 λ. Therefore, λi M BF ρ s RE = Y s j Ω Applying 3.3, λi M R{BF} becomes r Y s j λ j Ω D rnj nj+1 j s=1 D j λ nj nj+1 m j 1 i=1 j m j 1 i=1 D j,i λ rnj+1. D j,i λ nj On the Laplacian, signless Laplacian and adjacency eigenvalues of R{BF}. For each j, let and let µ 1 G j µ 2 G j µ mj 1G j µ mj G j = 0, µ 1 R µ 2 R µ r R = 0 be the Laplacian eigenvalues of G j and R, respectively. Applying Theorem 3.7 to the determination of the Laplacian spectrum of G, we obtain: 4.1 Theorem 4.1. The Laplacian spectrum of G = R{BF} is σlg = j Ω σu j rnj nj+1 j mj 1 i=1 σu j,i rnj+1 r s=1σw s

10 714 D.M. Cardoso, E.A. Martins, M. Robbiano, and O. Rojo where, for j = 1,...,k 1, U j is the j j leading principal submatrix of the k k matrix U k = X k except for the diagonal entries which in U k are δ 1,δ 2,...,δ k 1,δ k ; and, for i = 1,2,...,m j 1, U j,i = X j,i except for the diagonal entries which in U j,i are δ 1,δ 2,...,δ j 1,δ j +µ i G j ; and, for s = 1,2,...,r, W s = Y s except for the diagonal entries which in W s are δ 1,δ 2,...,δ k 1,δ k +µ s R. The multiplicities of the eigenvalues of LG are considered as in Theorem 3.7. Proof. It must be noted that, since M G = LG, we have a = 1, { δj I mj +LG j if j, S j = δ j I mj if j / and S k = δ k I r +LR. Therefore, LG j e mj = 0 = 0e mj and the Laplacian spectrum of G is given by Theorem 3.7, replacing the matrices X j, X j,i, and Y s by the matrices U j, U j,i and W s, respectively. As above, let G = R{BF}. We apply now Theorem 3.7 to find the eigenvalues of QG and AG whenever each G j is a regular graph of degree r j. For convenience, the signless Laplacian eigenvalues and adjacency eigenvalues are denoted in increasing order. Let and q 1 G q 2 G q 3 G q m 1 G q m G λ 1 G λ 2 G λ m 1 G λ m G be the eigenvalues of the signless Laplacian matrix and adjacency matrix of any graph G, respectively. If G is a regular graph of degree t and order m in which the edges have a weight equal to u then QGe m = 2tue m and AGe m = tue m. In this case, we may write λ m G = tu and q m G = 2tu. Theorem 4.2. If for each j the graph G j is a regular graph of degree r j and r j = 0 whenever j /, then the signless Laplacian spectrum of G = R{BF} is σqg = j Ω σv j rnj nj+1 j mj 1 i=1 σv j,i rnj+1 r s=1σu s

11 Copies of a Rooted Weighted Graph Attached to an Arbitrary Weighted Graph 715 where, for j = 1,2,3,...,k 1, V j is the j j leading principal submatrix of V k = X k except for the diagonal entries which in V k are δ 1 +2r 1 u 1,δ 2 +2r 2 u 2,...,δ k 1 +2r k 1 u k 1,δ k ; and, for i = 1,2,...,m j 1, V j,i = X j,i except for the diagonal entries which in V j,i are δ 1 +2r 1 u 1,δ 2 +2r 2 u 2,...,δ j 1 +2r j 1 u j 1,δ j +q i G j ; and, for s = 1,2,...,r, U s = Y s except for the diagonal entries which in U s are δ 1 +2r 1 u 1,δ 2 +2r 2 u 2,...,δ k 1 +2r k 1 u k 1,δ k +q s R. The multiplicities of the eigenvalues of QG must be considered as in Theorem 3.7. Proof. For M G = QG, we have a = 1, { δj I mj +QG j, S j = QG j = 0 if j / and S k = δ k I r +QR, QG j e mj = 2r j u j e mj if j and r j = 0 for j /. From Theorem 3.7, we obtain that the set of eigenvalues of QG is given replacing the matrices X j, X j,i, and Y s by the matrices V j, V j,i and U s, respectively. Theorem 4.3. If for each j the graph G j is a regular graph of degree r j and r j = 0 whenever j /, then the adjacency spectrum of G = R{BF} is σag = j Ω σt j rnj nj+1 j mj 1 i=1 σt j,i rnj+1 r s=1 σrs where, for j = 1,2,3,...,k 1, T j is the j j leading principal submatrix of T k = X k except for the diagonal entries which in T k are r 1 u 1,r 2 u 2,...,r k 1 u k 1,0; and, for i = 1,2,...,m j 1, T j,i = X j,i except for the diagonal entries which in T j,i are r 1 u 1,r 2 u 2,...,r j 1 u j 1,λ i G j ; and, for s = 1,2,...,r, Rs = Y s except for the diagonal entries which in Rs are r 1 u 1,r 2 u 2,...,r k 1 u k 1,λ s R. The multiplicities of the eigenvalues of QG must be considered as in Theorem 3.7.

12 716 D.M. Cardoso, E.A. Martins, M. Robbiano, and O. Rojo Proof. For M G = AG, we have a = 0, { AGj, S j = AG j = 0 if j / and S k = AR, and AG j e mj = r j e mj if j and r j = 0 for j /. Then, from Theorem 3.7, we conclude that the set of eigenvalues of AG is obtained replacing the matrices X j, X j,i, and Ys by the matrices T j, T j,i and Rs, respectively. 5. On the Randić eigenvalues of R{BF}. Let H be a simple connected graph with n vertices v 1,v 2,...,v n. Denote by dv 1,dv 2,...,dv n the degrees of v 1,v 2,...,v n, respectively. The Randić matrix of H is the square matrix of order n whose i,j entry is equal to 1 dvi dv j if v i and v j of H are connected and 0 otherwise [1]. The Randić eigenvalues of H are the eigenvaluesof the Randić matrix of H. The purpose of this section is to determine the Randić eigenvalues of G = R{BF} when each G j is regular of degree r j and R is a connected regular graph of degree p on r vertices v 1,v 2,...,v r. Keep in mind that r j = 0 if j /. As usual v i v j means that v i and v j are adjacent. Observe that, for i = 1,2,...,r, the degree of v k as a vertex of R{BF} is d k +p. The Randić matrix of R{BF} is the adjacency matrix of the weighted graph in which the edges joining the vertices at the level j+1 with the vertices at the level j of BF have weights 5.1 w k j = w k 1 = 1 dk j+1 +r k j+1 d k j +r k j 1 dk +pd k 1 +r k 1, the edges of graph G j have weights 2 j k 1, 5.2 u j = { 1 d j+r j if j, 0 if j /, and the weights of the edge v i v l of R are 5.3 ε i,l = ε l,i = { 1 p+d k if v i v l, 0 otherwise. Theorem 5.1. If for each j the graph G j is a regular graph of degree r j and the graph R is a regular graph of degree p then the Randić spectrum of G = R{BF}

13 Copies of a Rooted Weighted Graph Attached to an Arbitrary Weighted Graph 717 is j Ω σt j rnj nj+1 j mj 1 i=1 σt j,i rnj+1 r s=1 σrs in which the matrices T j,t j,i and Rs are those of Theorem 4.3 with the weights indicated in 5.1,5.2 and 5.3. The eigenvalues multiplicities must be considered as in Theorem 3.7. REFERENCES [1] S.B. Bozkurt, A.D. Güngör, and I. Gutman. Randić spectral radius and randić energy. MATCH Commun. Math. Comput. Chem., 64: , [2] D.M. Cardoso, D. Cvetković, P. Rowlinson, and S.K. Simić. A sharp lower bound for the least eigenvalue of the signless Laplacian of a non-bipartite graph. Linear Algebra Appl., 429: , [3] D.M. Cardoso, E.A. Martins, M. Robbiano, and V. Trevisan. Computing the Laplacian spectra of some graphs. Discrete Appl. Math., 160: , [4] D. Cvetković, P. Rowlinson, and S.K. Simić. Signless Laplacian of finite graphs. Linear Algebra Appl., 423: , [5] D. Cvetković and S.K. Simić. Towards a spectral theory of graphs based on the signless Laplacian, I. Publ. Inst. Math. Beograd, 8599:19 33, [6] D. Cvetković and S.K. Simić. Towards a spectral theory of graphs based on the signless Laplacian, II. Linear Algebra Appl., 432: , [7] D. Cvetković and S.K. Simić. Towards a spectral theory of graphs based on the signless Laplacian, III. Appl. Anal. Discrete Math., 4: , [8] R. Fernandes, H. Gomes, and E.A. Martins. On the spectra of some graphs like weighted rooted trees. Linear Algebra Appl., 428: , [9] M. Fiedler. Algebraic connectivity of graphs. Czechoslovak Math. J., 23: , [10] O. Rojo and L. Medina. Spectral characterization of some weighted rooted graphs with cliques. Linear Algebra Appl., 433: , [11] O. Rojo, M. Robbiano, D.M. Cardoso, and E.A. Martins. Spectra of weighted rooted graphs having prescribed subgraphs at some levels, Electron. J. Linear Algebra, 22: , [12] F. Zhang. Matrix Theory. Springer, New York, 1999.

Spectra and Randić Spectra of Caterpillar Graphs and Applications to the Energy

MATCH Communications in Mathematical and in Computer Chemistry MATCH Commun. Math. Comput. Chem. 77 07) 6-75 ISSN 0340-653 Spectra and Randić Spectra of Caterpillar Graphs and Applications to the Energy