Let Q(G), \({{\mathcal {D}}(G)}\) and \({{\mathcal {D}}}^Q(G)={{\mathcal {D}}iag(Tr)} + {{\mathcal {D}}(G)}\) be, respectively, the signless Laplacian matrix, the distance matrix and the distance signless Laplacian matrix of graph G, where \({{\mathcal {D}}iag(Tr)}\) denotes the diagonal matrix of the vertex transmissions in G. The eigenvalues of Q(G) and \({{\mathcal {D}}}^Q(G)\) will be denoted by \(q_{1} \ge q_{2} \ge \cdots \ge q_{n-1} \ge q_n\) and \(\partial ^Q_1 \ge \partial ^Q_2 \ge \cdots \ge \partial ^Q_{n-1} \ge \partial ^Q_n\) , respectively. A graph G which does not share its distance signless Laplacian spectrum with any other non-isomorphic graphs is said to be determined by its distance signless Laplacian spectrum. Characterizing graphs with respect to spectra of graph matrices is challenging. In literature, there are many graphs that are proved to be determined by the spectra of some graph matrices (adjacency matrix, Laplacian matrix, signless Laplacian matrix, distance matrix etc.). But there are much fewer graphs that are proved to be determined by the distance signless Laplacian spectrum. Namely, the path graph, the cycle graph, the complement of the path and the complement of the cycle are proved to be determined by the distance signless Laplacian spectra. In this paper, we establish Nordhaus–Gaddum-type results for the least signless Laplacian eigenvalue of graph G. Moreover, we prove that the join graph \(G\vee K_{q}\) is determined by the distance singless Laplacian spectrum when G is a \(p-2\) regular graph of order p. Finally, we show that the short kite graph and the complete split graph are determined by the distance signless Laplacian spectra. Our approach for characterizing these graphs with respect to distance signless Laplacian spectra is different from those given in literature.

设Q（G），\（{{\ mathcal {D}}（G）} \）和\（{{\ mathcal {D}}} ^ Q（G）= {{\ mathcal {D}} iag（ Tr）} + {{\数学{D}}（G）} \）分别是图G的无符号拉普拉斯矩阵，距离矩阵和距离无符号拉普拉斯矩阵，其中\（{{\ mathcal {D} } iag（Tr）} \）表示G中顶点传输的对角矩阵。的特征值Q（ģ）和\（{{\ mathcal {d}}} ^ Q（G）\）将通过被表示\（Q_ {1} \ GE Q_ {2} \ GE \ cdots \ GE Q_ { n-1} \ ge q_n \）和\（\ partial ^ Q_1 \ ge \ partial ^ Q_2 \ ge \ cdots \ ge \ partial ^ Q_ {n-1} \ ge \ partial ^ Q_n \）  。图G与其他非同构图没有共享其距离无符号拉普拉斯频谱的谱线据说由其距离无符号拉普拉斯谱确定。关于图矩阵的光谱表征图是具有挑战性的。在文献中，有许多图被证明是由某些图矩阵（邻接矩阵，拉普拉斯矩阵，无符号拉普拉斯矩阵，距离矩阵等）的光谱确定的。但事实证明，由距离无符号拉普拉斯频谱确定的图要少得多。即，路径图，循环图，路径的补码和循环的补码被证明是由距离无符号拉普拉斯谱确定的。在本文中，我们为图的最小无符号拉普拉斯特征值建立了Nordhaus–Gaddum型结果摹。此外，我们证明了连接图\（G \ VEE K_ {Q} \）由距离singless拉普拉斯谱确定当ģ是\（P-2 \）顺序的正则图p。最后，我们证明了短风筝图和完全分裂图是由距离无符号拉普拉斯光谱确定的。我们针对距离无符号拉普拉斯光谱描述这些图的方法与文献中给出的方法不同。