Spectra of the generalized edge corona of graphs

Transcription

1 Discrete Mathematics, Algorithms and Applications Vol 0, No 08) pages) c World Scientific Publishing Company DOI: 04/S Spectra of the generalized edge corona of graphs Yanyan Luo and Weigen Yan School of Sciences, Jimei University Xiamen 360, P R China @qqcom weigenyan@63net Received 3 December 06 Revised 7 September 07 Accepted 5 October 07 Published 4 November 07 Let G be a simple graph with m edges and H, H,,H m be m simple graphs The generalized edge corona, denoted by G[H i ] m, is the graph obtained by taking one copy of graphs G, H, H,,H m and then joining two end-vertices of the ith edge e i of G to every vertex of H i for i m In this paper, we determine and study the characteristic polynomial, Laplacian polynomial and signless Laplacian polynomial of G[H i ] m Asan application, we also count the number of spanning trees of the generalized edge corona Keywords: Edge corona; generalized edge corona; spanning tree Mathematics Subject Classification 00: 05C50, 05C05 Introduction Throughout this paper, I n and j n are identity matrix of order n and length-n column vector consisting entirely of s, respectively Let G V,E) be a graph with the vertex set V G) {v,v,,v n },theedgeseteg) {e,e,,e m }The adjacency matrix of G is an n n matrix AG) whosei, j)-entry is if v i is adjacent to v j and 0 otherwise The vertex-edge incidence matrix RG) r ij )ofg is an n m matrix with entry r ij if the vertex v i is incident to the edge e j and 0 otherwise The characteristic polynomial of G, denoted by f G λ), is the characteristic polynomial of AG) The Laplacian matrix of G and the signless Laplacian matrix of G are defined as LG) G) AG) andqg) G)+AG), respectively, where G) is the diagonal matrix whose diagonal entries are degree sequences of G We denote the Laplacian polynomial of G by f LG) µ) and the signless Laplacian polynomial of G by f QG) ν) Denote the eigenvalues of AG), LG) andqg), respectively, by λ λ λ n, 0µ <µ µ n,ν ν ν n Corresponding author

2 Y Luo & W Yan a) b) Fig a) A connected graph G with edge set {e,e,e 3 } and three graphs H,H and H 3 b) The generalized edge corona G[H,H,H 3 ] Harary [9] defined the corona of two graphs and Frucht and Harary [8] obtained some results on the corona of two graphs 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 [3] For some related results to the corona of graphs, see [, 5 7, 3] Hou and Shiu [0] defined the edge corona of two graphs G and H, denoted by G H, as follows Suppose G has m edges, e,e,,e m Takem copies of H, denoted by H,H,,H m, and for each edge e k u, v) ofg, adding edges between the two end-vertices u and v of e k and each vertex of the H k The resulting graph is G H Hou and Shiu [0] studied some of spectral properties Moreover, the edge corona of two graphs has been extensively studied by mathematicians For some related results, see [6,, 4] We now generalize the definition of edge corona of graphs Let G be a simple graph with n vertices and m edges, H, H,,H m be m simple graphs The generalized edge corona, denoted by G[H i ] m, is the graph obtained by taking one copy of graphs G, H, H,,H m and then joining two end-vertices of the ith edge e i of G to every vertex of H i for i m For example, let G, H,H and H 3 be the graphs illustrated in Fig a), then the graph G[H,H,H 3 ] can be illustrated in Fig b) Obviously, if H H H m H, theng[h i ] m G H In this paper, we give a complete description of the characteristic polynomial, Laplacian polynomial, and signless Laplacian polynomial of G[H i ] m As an application, we also count the number of spanning trees of the generalized edge corona Preliminaries The following result will play an important role in the proof of the main results Lemma []) Let A [ A A ] A A be a matrix, where A and A are square matrices If A and A are invertible, then A A det deta )deta A A A A A ) deta )deta A A A )

3 Spectra of the generalized edge corona of graphs Let A a ij )andb be two matrices Then the Kronecker product of A and B is defined to be the partition matrix a ij B) and is denoted by A B Let G be an r -regular graph with n vertices and m edges, and H, H,,H m be mr -regular graphs with n vertices Then the adjacency matrix AG[H i ] m )of G[H i ] m can be expressed by ) AG[H i ] m ) A R jn T, ) R T j n D where A : AG) is the adjacency matrix of G and R is the vertex-edge incidence matrix of G, and B B 0 D, 0 0 B m and B i : AH i ) is the adjacency matrix of H i, respectively, for i,,,msimilarly, let G be an r -regular graph with n vertices and m edges, and H, H,,H m be m simple graphs with n vertices Then the Laplacian matrix of G[H i ] m can be written as LG)+r n LG[H i ] m I n R jn T ) R T j n LH i m )+I, ) n m where LG) is the Laplacian matrix of G and LH ) LH LH i m ) 0 ) 0 0 LH m ) Similarly, let G be an r -regular graph with n vertices and m edges, and H, H,,H m be mr -regular graphs with n vertices Then the signless Laplacian matrix of G[H i ] m can be written as QG)+r QG[H i ] m ) n I n R jn T, 3) R T j n QH i m )+I nm where QG) is the signless Laplacian matrix of G and QH ) QH QH i m ) ) QH m )

4 Y Luo & W Yan Keep the notations above and set χ H λ) R jn T λi nm D) R T j n, χ LH) µ) R jn T µ )I nm LH i m )) R T j n, χ QH) ν) R jn T ν )I nm QH i m )) R T j n Proposition χ H λ) n λ r A + r I n ), 4) χ LH) µ) n µ A + r I n ), 5) χ QH) ν) n ν r A + r I n ) 6) Proof Note that each row sum of D equals r Hence DR T j n )r R T j n ), and so Thus, λi nm D)R T j n λ r )R T j n R jn T λi nm D) R T j n R jt n ) R T j n ) n A + r I n ), λ r λ r as required of the first result Similarly, we can prove the second and third ones 3 The Characteristic Polynomial of G[H i ] m Now, we can compute the characteristic polynomial of G[H i ] m Theorem Let G be an r -regular graph with n vertices and m edges, H, H,,H m be mr -regular graphs with n vertices Then the characteristic polynomial of G[H i ] m can be expressed by f G[Hi] m λ) λ r ) n f Hi λ) [λ λ j )λ r ) n r + λ j )], where λ λ λ n r are the eigenvalues of G Proof By Eq ), it follows that λin f G[Hi] mλ) det A R jn T R T j n λi nm D

5 Spectra of the generalized edge corona of graphs Using Lemma and Eq 4), one may obtain that f G[Hi] mλ) detλi n m D) detλi n A R jn T λi nm D) R T j n ) λi n B det λi n B det λi n A n ) A + r I n ) λ r f Hi λ) det f Hi λ) λ r ) n This completes the proof λ n r λ r λin B m ) I n + n ) ) A λ r [ λ λ j ) n ] r + λ j ) λ r f Hi λ) [λ λ j )λ r ) n r + λ j )] Note that G is an r -regular graph with n vertices and m edges, and H,H,,H m are mr -regular graphs with n vertices We may assume that the eigenvalues of H i are λ i) λ i) λ i) n r for i,,,m,andthe eigenvalues of G are λ λ λ n r Set α j, ᾱ j r + λ j ) ± r λ j ) +4n r + λ j ) The following result is immediate from the theorem above Corollary 3 Let G be an r -regular connected graph with n vertices and m edges, and H,H,,H m be m r -regular graphs with n vertices Then spectrum of G[H i ] m is { m n } {}}{ r,,r {λ j) i i,,,n,j,,,m} {α j, ᾱ j j,,,n} 4 The Laplacian Polynomial of G[H i ] m Now, we can obtain the expression of the Laplacian polynomial of G[H i ] m as follows Theorem 4 Let G be an r -regular graph with n vertices and m edges, H, H,,H m be m simple graphs with n vertices Then f LG[Hi] m ) µ) µ ) n f LHi)µ ) [µ )µ µ j ) n µr µ j )], where 0µ <µ µ n are the Laplacian eigenvalues of G

6 Y Luo & W Yan Proof By Eq ), it follows that µin LG) r n I n R jn T f LG[Hi] m ) µ) det R T j n µi nm LH i m ) I n m) By Lemma and Eq 5), one may obtain that f LG[Hi] m ) µ) µ )I n LH ) det µ )I n LH ) µ )In LH m ) det µ r n )I n LG) n ) µ A + r I n ) f LHi)µ ) det µ µn ) r I n n ) ) LG) µ µ f LHi)µ ) as required µ ) n f LHi)µ ) [ µ µ j ) n ] µ µr µ j ) [µ )µ µ j ) n µr µ j )], Assume that the Laplacian eigenvalues of H i are 0 µ i) <µ i) µ i) n for i,,,m, and the Laplacian eigenvalues of G are 0 µ <µ µ n Set β j, β j +n r + µ j ± + n r + µ j ) 4µ j + n ) Then the following corollary is immediate from the theorem above Corollary 5 Let G be an r -regular connected graph with n vertices and m edges, and H,H,,H m be m simple graphs with n vertices Then Laplacian spectrum of G[H i ] m is ) µ ) + µ ) n + µ m) + µ m) n + β β β n βn, m n where entries in the first row are the eigenvalues with the number of repetitions written below, respectively

7 Spectra of the generalized edge corona of graphs 5 The Signless Laplacian Polynomial of G[H i ] m In this section, we obtain the expression of the signless Laplacian polynomial of G[H i ] m Theorem 6 Let G be an r -regular graph with n vertices and m edges, H, H,,H m be mr -regular graphs, each of them has n vertices Then f QG[Hi] m ) ν) ν r ) n f QHi)ν ) [ν ν j )ν r ) n r ν r ) n ν j ], where ν ν ν n r are the signless Laplacian eigenvalues of G Proof By Eq 3), it follows that νin QG) r n I n R jn T f QG[Hi] m ) ν) det R T j n νi nm QH i m ) I nm By Lemma and Eq 6), one may obtain that f QG[Hi] m ) ν) ν )I n QH ) det ν )I n QH ) ν )In QH m ) ) ) n det νi n n r I n + QG) ν r f QHi)ν ) ν r ) n [ ν ν j ) f QHi)ν ) n ν r r ν r ) + ν j ) [ν ν j )ν r ) n r ν r ) n ν j ] ] Assume that the signless Laplacian eigenvalues of H i are ν i) ν i) ν n i) r for i,,,m, and the Laplacian eigenvalues of G are ν ν ν n r Seta +r + n r + ν j,and γ j, γ j a ± + r + n r + ν j ) 4[ν j + r n )+n r r +)] Then the following corollary is immediate from the theorem above

8 Y Luo & W Yan Corollary 7 Let G be an r -regular connected graph with n vertices and m edges, and H,H,, H m be mr -regular graphs with n vertices Then signless Laplacian spectrum of G[H i ] m is r + ν ) + ν ) n + νm) + ν m) n + γ ) γ γ n γ n, m n where entries in the first row are the eigenvalues with the number of repetitions written below, respectively 6 The Number of Spanning Trees of Generalized Edge Corona of Graphs As an application of Corollary 5, in this section, we obtain the formula of the number of spanning trees of the generalized edge corona of G[H i ] m Let G be a connected graph with n vertices and Laplacian eigenvalues 0 µ < µ µ n It is well known [4] that the number of spanning trees of G can be expressed by tg) µ µ 3 µ n 7) n Lemma 8 Let H be a graph with n vertices and 0µ <µ µ n be the Laplacian eigenvalues of H Then the number of spanning trees of K H can be expressed by tk H) n +)µ +)µ 3 +) µ n +), where K is the complete graph with two vertices Proof In fact, K H is the join K H or by Theorem 4), hence the Laplacian spectrum of K H is Hence, by Eq 7), as required 0, +n, +n,µ +,µ 3 +,,µ n + tk H) n +)µ +)µ 3 +) µ n +), Theorem 9 Let G be an r -regular connected graph with n vertices and m edges, H,H,, H m be m simple graphs with n vertices Then the number of spanning tree of G[H i ] m m n+ tg[h i ] m ) tg) tk H i ) n

9 Spectra of the generalized edge corona of graphs Proof By Corollary 5, the Laplacian spectrum of G[H i ] m is ) ) µ + µ ) n + µ m) + µ m) n + β β β n βn m n Note that β j βj +n )µ j for j,,n,andβ r n +, β 0Thus, tg[h i ] m ) m n r n +)n +) n n i µ) i +) n i µ) i +) n i µm) i +) n j µ j n + mn m n ntg)r n +)n +) n n i µ) i +) n i µ) i +) n i µm) i +) n + mn n n n m n+ tg)n +) n µ ) i +) µ ) i +) µ m) i +) i The last equality follows from n + n m n+rn) From Lemma 8, the theorem is immediate Acknowledgments We are grateful to the anonymous referees for many friendly and helpful revising suggestions that greatly improved the presentation of the paper This work is supported by NSFC Grant5739) References [] C Adiga and B R Rakshith, On spectra of variants of the corona of two graphs and some new equienergetic graphs, Discuss Math Graph Theory 36 06) 7 40 [] R B Bapat, Graphs and Matrices Springer, 00) [3] S Barik, S Pati and B K Sarma, The spectrum of the corona of two graphs, SIAM J Discrete Math 4 007) [4] N L Biggs, Algebraic Graph Theory, nd edn Cambridge University Press, Cambridge, 993) [5] C J Bu, J Zhou and H B Li, Spectral characterization of the corona of a cyle and two isolated vertices, Graphs Combin 30 04) 3 33 [6] S Y Cui and G-X Tian, The signless Laplacian spectrum of the edge) corona of two graphs, Util Math 88 0) [7] A R Fiuj Laali, H Haj Seyyed Javadi and D Kiani, Spectra of generalized corona of graphs, Linear Algebra Appl ) 4 45 [8] R Frucht and F Harary, On the corona of two graphs, Aequationes Math 4 970) 3 35 [9] F Harary, Graph Theory Addition-Wesley Publishing Co, Reading, CA/London, 969) [0] Y P Hou and W-C Shiu, The spectrum of the edge corona of two graphs, Electron J Linear Algebra 0 00) i i

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