单圈图的Seidel无符号拉普拉斯能量
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摘要
设G是一个具有n个顶点、m条边的简单图,S(G)表示G的Seidel矩阵,d_i表示顶点v_i的度,又以DS(G)=diag(n-1-2d_1,n-1-2d_2,…,n-1-2d_n)来表示对角矩阵,再依次定义图G的Seidel拉普拉斯矩阵为SL(G)=DS(G)-S(G)、图G的Seidel无符号拉普拉斯矩阵为SL~+(G)=DS(G)+S(G)和图G的Seidel无符号拉普拉斯能量为■,这里σ1L+,σ2L+,…,σnL+为矩阵SL+(G)的特征值.文章利用不等式讨论单圈图G的Seidel无符号拉普拉斯能量的上界,得到了几个有意义的结果.
Let G be a simple graph with n vertices and m edges,S(G)denotes the Seidel matrix of G,and let DS(G)=diag(n-1-2 d_1,n-1-2 d_2,…,n-1-2 d_n)be the diagonal matrix with d_i denoting the degree of a vertex v_i in G.The Seidel Laplacian matrix of G is defined as SL(G)=DS(G)-S(G)and the Seidel signless Laplacian matrix as SL~+(G)=DS(G)+S(G),and the Seidel signless Laplacian energy as ■,there,σ1 L+,σ2 L+,…,σnL+ are the eigenvalues of SL+(G).By using some inequality,we discuss the upper bounds of the Seidel signless Laplacian energy of unicyclic graphs,and obtain some significant conclusions.
Let G be a simple graph with n vertices and m edges,S(G)denotes the Seidel matrix of G,and let DS(G)=diag(n-1-2 d_1,n-1-2 d_2,…,n-1-2 d_n)be the diagonal matrix with d_i denoting the degree of a vertex v_i in G.The Seidel Laplacian matrix of G is defined as SL(G)=DS(G)-S(G)and the Seidel signless Laplacian matrix as SL~+(G)=DS(G)+S(G),and the Seidel signless Laplacian energy as ■,there,σ1 L+,σ2 L+,…,σnL+ are the eigenvalues of SL+(G).By using some inequality,we discuss the upper bounds of the Seidel signless Laplacian energy of unicyclic graphs,and obtain some significant conclusions.
引文
[1]LI X L,SHI Y T,GUTMAN I.Graph energy[M].New York:Springer,2012.
[2]GUTMAN I,FURTULA B.Survey of graph energies[J].Mathematics Interdisciplinary Research,2017,2(2):85-129.
[3]GUTMAN I,ZHOU B.Laplacian energy of a graph[J].Linear Algebra and its Applications,2006,414(1):29-37.
[4]DAS K C,MOJALLAL S A.On Laplacian energy of graphs[J].Discrete Mathematics,2014,325:52-64.
[5]DAS K C,MOJALLAL S A,GUTMAN I.On Laplacian energy in terms of graph invariants[J].Applied Mathematics and Computation,2015,268:83-92.
[6]ABREU N,CARDOSO D M,GUTMAN I,et al.Bounds for the signless Laplacian energy[J].Linear Algebra and its Applications,2011,435(10):2365-2374.
[7]HAEMERS W H.Seidel switching and graph energy[J].MATCH Communications in Mathematical and in Computer Chemistry,2012,68:653-659.
[8]NAGESWARI P,SARASIJA P B.Seidel energy and its bounds[J].International Journal of Mathematical Analysis,2014,8:2869-2871.
[9]OBOUDI M R.Energy and Seidel energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2016,75:291-303.
[10]RAMANE H S,JUMMANNAVER R B,GUTMAN I.Seidel Laplacian energy of graphs[J].International Journal of Applied Graph Theory,2017,1(2):74-82.
[11]CVETKOVIC D,ROWLINSON P,SIMIC S K.Signless Laplacians of finite graphs[J].Linear Algebra and its Applications,2007,423(1):155-171.
[12]GOLUB G H,VAN LOAN C F.Matrix computations:4th edition[M].Maryland:Johns Hopkins University Press,2013.
[13]RAMANE H S,GUTMAN I,PATIL J B,et al.Seidel signless Laplacian energy of graphs[J].Mathematics Interdisciplinary Research,2017,2(2):181-191.
[2]GUTMAN I,FURTULA B.Survey of graph energies[J].Mathematics Interdisciplinary Research,2017,2(2):85-129.
[3]GUTMAN I,ZHOU B.Laplacian energy of a graph[J].Linear Algebra and its Applications,2006,414(1):29-37.
[4]DAS K C,MOJALLAL S A.On Laplacian energy of graphs[J].Discrete Mathematics,2014,325:52-64.
[5]DAS K C,MOJALLAL S A,GUTMAN I.On Laplacian energy in terms of graph invariants[J].Applied Mathematics and Computation,2015,268:83-92.
[6]ABREU N,CARDOSO D M,GUTMAN I,et al.Bounds for the signless Laplacian energy[J].Linear Algebra and its Applications,2011,435(10):2365-2374.
[7]HAEMERS W H.Seidel switching and graph energy[J].MATCH Communications in Mathematical and in Computer Chemistry,2012,68:653-659.
[8]NAGESWARI P,SARASIJA P B.Seidel energy and its bounds[J].International Journal of Mathematical Analysis,2014,8:2869-2871.
[9]OBOUDI M R.Energy and Seidel energy of graphs[J].MATCH Communications in Mathematical and in Computer Chemistry,2016,75:291-303.
[10]RAMANE H S,JUMMANNAVER R B,GUTMAN I.Seidel Laplacian energy of graphs[J].International Journal of Applied Graph Theory,2017,1(2):74-82.
[11]CVETKOVIC D,ROWLINSON P,SIMIC S K.Signless Laplacians of finite graphs[J].Linear Algebra and its Applications,2007,423(1):155-171.
[12]GOLUB G H,VAN LOAN C F.Matrix computations:4th edition[M].Maryland:Johns Hopkins University Press,2013.
[13]RAMANE H S,GUTMAN I,PATIL J B,et al.Seidel signless Laplacian energy of graphs[J].Mathematics Interdisciplinary Research,2017,2(2):181-191.