On the distance signless Laplacian of a graph

The distance signless Laplacian of a connected graph[GRAPHICS]is defined by[GRAPHICS], where[GRAPHICS]is the distance matrix of[GRAPHICS], and[GRAPHICS]is the diagonal matrix whose main entries are the vertex transmissions in[GRAPHICS]. The spectrum of[GRAPHICS]is called the distance signless Laplacian spectrum of[GRAPHICS]. In the present paper, we study some properties of the distance signless Laplacian eigenvalues. Among other results, we show that the complete graph is the unique graph with only two distinct distance signless Laplacian eigenvalues. We prove several bounds on[GRAPHICS]eigenvalues and establish a relationship between[GRAPHICS]being a distance signless Laplacian eigenvalue of[GRAPHICS]and[GRAPHICS]containing a bipartite component.

连通图[图形]的距离无符号拉普拉斯算子由[图形]定义，其中[图形]是[图形]的距离矩阵，[图形]是对角矩阵，其主对角元是[图形]中的顶点传递度。[图形]的谱称为[图形]的距离无符号拉普拉斯谱。在本文中，我们研究距离无符号拉普拉斯特征值的一些性质。除其他结果外，我们表明完全图是唯一只有两个不同距离无符号拉普拉斯特征值的图。我们证明了[图形]特征值的几个界，并建立了[图形]是[图形]的距离无符号拉普拉斯特征值与[图形]包含一个二部图分量之间的关系。