On (distance) signless Laplacian spectra of graphs

Let Q(G), D(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {D}}(G)}$$\end{document} and DQ(G)=Diag(Tr)+D(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {D}}}^Q(G)={{\mathcal {D}}iag(Tr)} + {{\mathcal {D}}(G)}$$\end{document} be, respectively, the signless Laplacian matrix, the distance matrix and the distance signless Laplacian matrix of graph G, where Diag(Tr)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {D}}iag(Tr)}$$\end{document} denotes the diagonal matrix of the vertex transmissions in G. The eigenvalues of Q(G) and DQ(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathcal {D}}}^Q(G)$$\end{document} will be denoted by q1≥q2≥⋯≥qn-1≥qn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q_{1} \ge q_{2} \ge \cdots \ge q_{n-1} \ge q_n$$\end{document} and ∂1Q≥∂2Q≥⋯≥∂n-1Q≥∂nQ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial ^Q_1 \ge \partial ^Q_2 \ge \cdots \ge \partial ^Q_{n-1} \ge \partial ^Q_n$$\end{document} , 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∨Kq\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$G\vee K_{q}$$\end{document} is determined by the distance singless Laplacian spectrum when G is a p-2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p-2$$\end{document} 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.