The spectrum of the edge corona of two graphs

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1 Electronic Journal of Linear Algebra Volume Volume (1) Article 4 1 The spectrum of the edge corona of two graphs Yaoping Hou yphou@hunnu.edu.cn Wai-Chee Shiu Follow this and additional works at: Recommended Citation Hou, Yaoping and Shiu, Wai-Chee. (1), "The spectrum of the edge corona of two graphs", Electronic Journal of Linear Algebra, Volume. 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 Volume, pp , September 1 THE SPECTRUM OF THE EDGE CORONA OF TWO GRAPHS YAOPING HOU AND WAI-CHEE SHIU Abstract. Given two graphs, with vertices 1,,..., n and edges e 1, e,..., e m, and G, the edge corona G of and G is defined as the graph obtained by taking m copies of G and for each edge e k = ij of G, joining edges between the two end-vertices i, j of e k and each vertex of the k-copy of G. In this paper, the adjacency spectrum and Laplacian spectrum of G are given in terms of the spectrum and Laplacian spectrum of and G, respectively. As an application of these results, the number of spanning trees of the edge corona is also considered. Key words. Spectrum, Adjacency matrix, Laplacian matrix, Corona of graphs. AMS subject classifications. 5C5, 5C5. 1. Introduction. Throughout this paper, we consider only simple graphs. Let G = (V,E) be a graph with vertex set V = {1,,...,n}. The adjacency matrix of G denoted by A(G) is defined as A(G) = (a ij ), where a ij = 1 if i and j are adjacent in G, otherwise. The spectrum of G is defined as σ(g) = (λ 1 (G),λ (G),...,λ n (G)), where λ 1 (G) λ (G)... λ n (G) are the eigenvalues of A(G). The Laplacian matrix of G, denoted by L(G) is defined as D(G) A(G), where D(G) is the diagonal degree matrix of G. The Laplacian spectrum of G is defined as S(G) = (θ 1 (G),θ (G),...,θ n (G)), where = θ 1 (G) θ (G)... θ n (G) are the eigenvalues of L(G). We call λ n (G) and θ n (G) the spectral radius and Laplacian spectral radius, respectively. There is extensive literature available on works related to spectrum and Laplacian spectrum of a graph. See [, 5, 6] and the references therein to know more. The corona of two graphs is defined in [4] and there have been some results on the corona of two graphs [3]. The complete information about the spectrum of the corona of two graphs G,H in terms of the spectrum of G,H are given in [1]. In this Received by the editors September 11, 9. Accepted for publication August, 1. Handling Editor: Richard A. Brualdi. Department of Mathematics, Hunan Normal University, Changsha, Hunan 4181, China (yphou@hunnu.edu.cn). Research supported by NSFC(177161). Department of Mathematics, Hong Kong Baptist University, Hong Kong, China. (wcshiu@hkbu.edu.hk). 586

3 Volume, pp , September 1 The Spectrum of the Edge Corona of Two Graphs 587 paper, we consider a variation of the corona of two graphs and discuss its spectrum and the number of spanning trees. Definition 1.1. Let and G be two graphs on disjoint sets of n 1 and n vertices, m 1 and m edges, respectively. The edge corona G of and G is defined as the graph obtained by taking one copy of and m 1 copies of G, and then joining two end-vertices of the i-th edge of to every vertex in the i-th copy of G. Note that the edge corona G of and G has n 1 + m 1 n vertices and m 1 + m 1 n + m 1 m edges. Example 1.. Let be the cycle of order 4 and G be the complete graph K of order. The two edge coronas G and G are depicted in Figure G G G Figure 1: An example of edge corona graphs Throughout this paper, is assumed to be a connected graph with at least one edge. In this paper, we give a complete description of the eigenvalues and the corresponding eigenvectors of the adjacency matrix of G when and G are both regular graphs and give a complete description of the eigenvalues and the corresponding eigenvectors of the Laplacian matrix of G for a regular graph and arbitrary graph G. As an application of these results, we also consider the number of spanning trees of the edge corona.. The spectrum of the graph G. Let the vertex set and edge set of a graph G be V = {1,,...,n} and E = {e 1,e,...,e m }, respectively. The vertex-edge incidence matrix R(G) = (r ij ) is an n m matrix with entry r ij = 1 if the vertex i is incident the edge e j and otherwise. Lemma.1. [, P. 114] Let G be a connected graph with n vertices and R be the vertex-edge incident matrix. Then rank(r) = n 1 if G is bipartite and n otherwise. Lemma.. [] Let G be a connected graph with spectral radius ρ. Then ρ is also an eigenvalue of A(G) if and only if G is bipartite. Moreover, if G is a

4 Volume, pp , September Y. Hou and W. Shiu connected bipartite graph with vertex partition V = V 1 V and X = (X 1,X ) T is an eigenvector corresponding eigenvalue λ of A(G) then X = (X 1, X ) T is an eigenvector corresponding eigenvalue λ of A(G). Let A = (a ij ),B be matrices. Then the Kronecker product of A and B is defined the partition matrix (a ij B) and is denoted by A B. The row vector of size n with all entries equal to one is denoted by j n and the identity matrix of order n is denoted by I n. Let and G be graphs with n 1,n vertices and m 1,m edges,respectively. Then the adjacency matrix of G = G can be written as A(G) = A( ) R( ) j n (R( ) j n ) T I m1 A(G ) where A( ) and A(G ) are the adjacency matrices of the graphs and G, respectively, and R( ) is the vertex-edge incidence matrix of. A complete characterization of the eigenvalues and eigenvectors of G will be given when both and G are regular., Let be an r 1 -regular graph and G be an r -regular graph and (.1) σ( ) = (µ 1,µ,...,µ n1 ), σ(g ) = (η 1,η,...,η n ) be their adjacency spectrum, respectively. If is 1-regular then = K as is connected. In this case, G is the complete product of K and G. By Theorem.8 of [], or by some direct computations, we can obtain the spectrum of G = G as (η 1,...,η n 1,µ 1 = r+µ1 (r µ 1) +4(r 1+µ 1)n, r+µ± (r µ ) +4(r 1+µ )n ), where µ 1 = 1,µ = 1 are the spectrum of K. Theorem.3. Let be an r 1 -regular (r 1 ) graph and G be an r -regular graph and their spectra are as in (.1). Then the spectrum σ(g) of G is ( η 1 η η n = r r +µ 1± (r µ 1) +4(r 1+µ 1)n m 1 m 1 m 1 n 1 1 r +µ n1 ± (r µ n1 ) +4(r 1+µ n1 )n 1 ) where entries in the first row are the eigenvalues with the number of repetitions written below, respectively.

5 Volume, pp , September 1 The Spectrum of the Edge Corona of Two Graphs 589 G Z j e i G X i s t u v G X i (s)+x i (t) λ i r j G X i (u)+x i (v) λ i r j s t u v Y stj G G Y uvj G Figure : Description of adjacency eigenvectors Proof. Let Z 1,Z,...,Z n be the orthogonal eigenvectors of A(G ) corresponding to the eigenvalue η 1,η,..., η n = r, respectively. Note that G is r -regular and Z j j for j = 1,,...,n 1. Then for i = 1,,...,m 1 and for j = 1,,...,n 1, we have (see Figure, picture on the left) that (n 1 + m 1 n )-dimension vectors (,,...,,Z j,,...,) T, where (i+1)-th block is Z j are eigenvectors of G corresponding to eigenvalue η j. Thus we obtain m 1 (n 1) eigenvalues and corresponding eigenvectors of G. Let X 1,X,...,X n1 be the orthogonal eigenvectors of A( ) corresponding to the eigenvalues µ 1,µ,...µ n1, respectively. For i = 1,,...,n 1, let and λ i = r + µ ni + (r µ ni ) + 4(r 1 + µ ni )n λ i = r + µ ni (r µ ni ) + 4(r 1 + µ n )n. Note that r+µn i ± (r µ ni ) +4(r 1+µ ni )n = r if and only if µ i = r 1. So λ i or λ i is r if and only if is bipartite (note that at most one of λ i is r ). If is bipartite and the bipartition of its vertex set is V 1 V, then by Lemma. and some computations, we obtain that (j, j,,...,) T (1 on V 1, 1 on V, and on all copies of G ) is an eigenvector of G corresponding the eigenvalue r 1. Observe that if λ i and λ i are not equal to r then λ i and λ i are eigenvalues of G corresponding to the eigenvectors F i = (X i,..., Xi(s)+Xi(t) λ i r,...) T and F i = (X i,..., Xi(s)+Xi(t),...) T, respectively (see Figure, picture in the middle). In fact, it λ i r needs only to be checked that characteristic equations v u F i(v) = λ i F i (u) (resp. F v u i (v) = λ i Fi (u)) hold for every vertex u in G. For any vertex u in k-copy of G, let edge e k = st, then F i (u) = Xi(s)+Xi(t) λ i r. Furthermore, F i (v) = r F i (u) + X i (s) + X i (t) = λ i F i (u). v u

6 Volume, pp , September 1 59 Y. Hou and W. Shiu For any vertex u in, F i (v) = v u v u,v V () F i (v) + v u,v V () = µ i X i (u) + r 1n X i (u) λ i r + n λ i r = λ i X i (u) = λ i F i (u). F i (v) v u,v V () F i (v) Therefore we obtain n 1 eigenvalues and corresponding eigenvectors of G if is not bipartite and n 1 1 eigenvalues and corresponding eigenvectors of G if is bipartite. Let Y 1,Y,...,Y b be a maximal set of independent solution vectors of linear system R( )Y =. Then b = m 1 n 1 if is not bipartite and b = m 1 n if is bipartite. For i = 1,,...,b, let H i = (,Y i (e 1 )j,...,y i (e m )j) T (see Figure, picture on the right). We can obtain that H i is an eigenvector of G corresponding to eigenvalues r = η n. Thus these Y i s provide b eigenvalues and corresponding eigenvectors of G. Therefore we obtain n 1 + m 1 n eigenvalues and corresponding eigenvectors of G and it is easy to see that these eigenvectors of G are linearly independent. Hence the proof is completed. Next we consider the Laplacian spectrum of G. Let L( ) and L(G ) be the Laplacian matrices of the graphs and G, respectively, and R( ) be the vertex-edge incidence matrix of. Then the Laplacian matrix of G = G is L(G) = L( ) + r 1 n I n1 R( ) j n (R( ) j n ) T I m1 (I n + L(G )). In the following, we give a complete characterization of the Laplacian eigenvalues and eigenvectors of G. Let be an r 1 -regular graph and G be any graph and (.) S( ) = (θ 1,θ,...,θ n1 ), S(G ) = (τ 1,τ,...,τ n ) be their Laplacian spectra, respectively. If is 1-regular then = K as is connected. In this case, G is the complete product of K and G (by [5]), or by some direct computations, we can obtain that the Laplacian spectrum of G = G

7 Volume, pp , September 1 The Spectrum of the Edge Corona of Two Graphs 591 is S(G) = (,τ +,...,τ n +,n +,n + ). Theorem.4. Let be an r 1 -regular (r 1 ) graph and G be any graph and their Laplacian spectra are written as in (.). Let β i, β i = r 1n + θ i + ± (r 1 n + θ i + ) 4(n + )θ i for every θ i. Then the Laplacian spectrum S(G) of G is ( τ1 +, τ +,, τ n +, β 1, β1,, β n1, βn1 m 1 n 1 m 1 m ) where entries in the first row are the eigenvalues with the number of repetitions written below, respectively. G Z j e i G X i s t u v G X i (s)+x i (t) β i j X G i (u)+x i (v) β i j s t u v G G Y st j Y uv j G Figure 3: Description of Laplacian eigenvectors Proof. Let Z 1,Z,...,Z n be the eigenvectors of L(G ) corresponding to the eigenvalues = τ 1,τ,...,τ n. Note that Z j j for j =,...,n. Then for i = 1,,...,m 1 and for j =,3,...,n, we have that (n 1 +m 1 n )-dimension vectors (,,...,,Z j,,...,) T, where (i+1)-th block is Z j are eigenvectors of L(G) corresponding to eigenvalue τ j + (see Figure 3, picture on the left). Thus we obtain m 1 (n 1) eigenvalues and corresponding eigenvectors of L(G). Let X 1,X,...,X n1 be the orthogonal eigenvectors of L( ) corresponding to the eigenvalues θ 1,θ,...,θ n1, respectively. For i = 1,,...,n 1, note that: β i, β i = r1n+θi+± (r 1n +θ i+) 4(n +)θ i = r1n+θi+± (r 1n +θ i ) +4n (r 1 θ i) since r 1,n 1, β i. Note that θ i r 1 and the equality holds if and only if is bipartite. Note that β i = implies that θ i = r 1. That is, βi = appears only if is bipartite and i = n 1. Moreover, if is bipartite and the bipartition of its vertex set is V 1 V, then it is easy to check that (j, j,,...,) T (1 on V 1,

8 Volume, pp , September 1 59 Y. Hou and W. Shiu 1 on V, and on all copies of G ) is an eigenvector corresponding the eigenvalue (n 1 + )r 1 = β n1 of L(G). Observe that if β i and β i are not equal to, then β i and β i are eigenvalues of L(G) and F i = (X i,..., Xi(s)+Xi(t) β i,...) T and F i = (X i,..., Xi(s)+Xi(t),...) T are eigenvectors β i of β i and β i respectively (see Figure 3, picture in the middle). In fact, it needs only to be checked that characteristic equations d G (u)f i (u) v u F i(v) = β i F i (u) (resp. d G (u) F i (u) F v u i (v) = β i Fi (u)) hold for every vertex u in G, where d G (u) is the degree of the vertex u in G. For every vertex u in k-copy of G, let the edge e k = st, then d G (u) = d G (u)+ and F i (u) = Xi(s)+Xi(t) β i. Further, d G (u)f i (u) v u F i (v) = d G (u) + )F i (u) d G (u) X i(s) + X i (t) β i (X i (s) + X i (t)) = β i F i (u). For every vertex u in, note that r 1 X i (u) v u X i (v) = θ i X i (u). v V ( ) We have d G (u)f i (u) v u F i (v) = (r 1 + r 1 n )F i (u) = (r 1 + r 1 n )X i (u) v u,v V () X i (v) v u,v V () v u,v V () = (r 1 + r 1 n )( β i ) n r 1 + n θ i β i X i (u) + (θ i r 1 )X i (u) = β i X i (u) = β i F i (u). F i (v) + v u,v V () n β i (X i (u) + X i (v)) F i (v) Therefore we obtain n 1 eigenvalues and corresponding eigenvectors of L(G) if is not bipartite, and n 1 1 eigenvalues and corresponding eigenvectors of L(G) if is bipartite. Let Y 1,Y,...,Y b be a maximal set of independent solution vectors of the linear system R( )Y =. Then b = m 1 n 1 if is not bipartite, and b = m 1 n if is bipartite. For i = 1,,...,b, let H i = (,Y i (e 1 )j,...,y i (e m )j) T (see Figure 3, picture on the right). We can obtain that H i is an eigenvector corresponding the eigenvalue (= τ 1 + ) of L(G). Thus these Y i s provide b eigenvalues and corresponding eigenvectors of L(G).

9 Volume, pp , September 1 The Spectrum of the Edge Corona of Two Graphs 593 Therefore we obtain n 1 + m 1 n eigenvalues and corresponding eigenvectors of L(G) and it is easy to see that these eigenvectors of L(G) are linearly independent. Hence the proof is completed. As an application of the above results, we give the number of spanning trees of the edge corona of two graphs. Let G be a connected graph with n vertices and Laplacian eigenvalues = θ 1 < θ θ n. Then the number of spanning trees of G is By Theorem.4 we have t(g) = θ θ 3 θ n. n Proposition.5. For a connected r 1 -regular graph and arbitrary graph G, let the number of spanning trees of be t( ) and the Laplacian spectra of G be = τ 1 τ τ n. Then the number of spanning trees of G is t( G ) = m1 n1+1 (n + ) n1 1 t( )(τ + ) m1 (τ n + ) m1. Proof. Following the notions in Theorem.4, note that β i βi = (n + )θ i for i = 1,,...,n 1 and β 1 = r 1 n +, β 1 =. Thus t( G ) = m1 n1 (r 1 n + )(n + ) n1 1 n i= (τ i + ) m1 n 1 j= θ j n 1 + m 1 n = n 1 m1 n1 (r 1 n + )(n + ) n1 1 t( ) n i= (τ i + ) m1 n 1 + m 1 n = m1 n1+1 (n + ) n1 1 t( )(τ + ) m1 (τ n + ) m1. The last equality follows from n 1 + m 1 n = n1(+r1n). By Proposition.5, we have t(g K 1 ) = m n+1 3 n 1 t(g) for a regular graph G. In fact t(g K 1 ) = m n+1 3 n 1 t(g) holds for arbitrary graph G by the following proposition. Proposition.6. Let G be a connected graph with n vertices and m edges. Then the number of spanning trees of G K 1 is m n+1 3 n 1 t(g), where t(g) is the number of spanning trees of G. Proof. Note that the Laplacian matrix of G K 1 is ( L(G) + D(G) R L(G K 1 ) = R T I m ).

10 Volume, pp , September Y. Hou and W. Shiu Let (L(G)) 11 be the reduced Laplacian matrix of G obtained by removing the first row and first column of L(G) and R 1 be the matrix obtained by removing the first row of the vertex-edge incidence matrix R. By the Matrix-Tree theorem [], we have ( ) (L(G) + D(G))11 R t(g K 1 ) = det(l(g K 1 )) 11 = det 1 R1 T I m = m det[(l(g) + D(G)) 11 1 R 1R T 1 ], since RR T = D(G) + A(G), R 1 R T 1 = (D(G) + A(G)) 11. Thus t(g K 1 ) = m det( 3 (D(G) A(G)) 11 = m n+1 3 n 1 t(g). Acknowledgments. The authors would like to express their sincere gratitude to the referee for a very careful reading of the paper and for all his or her insightful comments and valuable suggestions, which made a number of improvements on this paper REFERENCES [1] S. Barik, S. Pati, and B. K. Sarma. The spectrum of the corona of two graphs. SIAM. J. Discrete Math., 4:47 56, 7. [] D. Cvetković, M. Doob, and H. Sachs. Spectra of Graphs: Theory and Application, Johann Ambrosius Barth Verlag, Heidelberg-Leipzig, [3] R. Frucht and F. Harary. On the corona two graphs. Aequationes Math., 4:3 35, 197. [4] F. Harary. Graph Theory. Addition-Wesley Publishing Co., Reading, MA/Menlo Park, CA/London, [5] R. Merris. Laplacian matrices of graphs: a survey. Linear Algebra Appl., 197/198: , [6] B. Mohar. The Laplacian spectrum of graphs, Graph Theory, Combinatorics and Applications. John Wiley, New York, pp , 1991.