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).

Key words： Distance signless Laplacian; spectral radius; bound

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