Distance signless Laplacian spectrum of a graph

Abstract

Let G be a simple connected graph with n vertices. The transmission T_v of a vertex v is defined to be the sum of the distances from v to all other vertices in G, that is, T_v = Σ_u∈V d_uv, where d_uv denotes the distance between u and v. Let T₁, ..., T_n be the transmission sequence of G. Let D = (d_ij)_n×n be the distance matrix of G, and \mathfrak{T} be the transmission diagonal matrix diag(T₁, ..., T_n). The matrix $$\mathcal{Q}(G)=\mathcal{T}+\mathcal{D}$$ is called the distance signless Laplacian of G. In this paper, we provide the distance signless Laplacian spectrum of complete k-partite graph, and give some sharp lower and upper bounds on the distance signless Laplacian spectral radius q(G).

Keywords

Distance signless Laplacian / spectral radius / bound

References

[1] Aouchiche M, Hansen P. Two Laplacians for the distance matrix of a graph. Linear Algebra Appl, 2013, 439: 21- 33

[2] Chen Y Y, Lin H Q, Shu J L. Sharp upper bounds on the distance spectral radius of a graph. Linear Algebra Appl, 2013, 439: 2659- 2666

[3] He C X, Liu Y, Zhao Z H. Some new sharp bounds on the distance spectral radius of graph. MATCH Commun Math Comput Chem, 2010, 63: 783- 788

[4] Indulal G. Sharp bounds on the distance spectral radius and the distance energy of graphs. Linear Algebra Appl, 2009, 430: 106- 113

[5] Lin H Q, Hong Y, Wang J F, Shu J L. On the distance spectrum of graphs. Linear Algebra Appl, 2013, 439: 1662- 1669

[6] Lin H Q, Shu J L. Sharp bounds on distance spectral radius of graphs. Linear Multilinear Algebra, 2013, 61 (4): 442- 447

[7] Lin H Q, Xue J, Shu J L. On the D_α-spectra of graphs. Linear Multilinear Algebra, 2019, 69 (6): 997- 1019

[8] Minc H. Nonnegative Matrices. New York: John Wiley & Sons Inc, 1988

[9] Shu J L, Wu Y R. Sharp upper bounds on the spectral radius of graphs. Linear Algebra Appl, 2004, 377: 241- 248

[10] Yu G L. On the least distance eigenvalue of a graph. Linear Algebra Appl, 2013, 439: 2428- 2433

[11] Yu G L, Wu Y R, Shu J L. Sharp bounds on the signless Laplacian spectral radii of graphs. Linear Algebra Appl, 2011, 434: 683- 687

RIGHTS & PERMISSIONS

2022 Higher Education Press