University of Isfahan Transactions on Combinatorics 2251-8657 10 1 2021 03 01 Some inequalities involving the distance signless Laplacian eigenvalues of graphs 9 29 24869 10.22108/toc.2020.121940.1715 EN Abdollah Alhevaz Faculty of Mathematical Sciences, Shahrood University of Technology, P. O. Box: 316-3619995161, Shahrood, Iran 0000-0001-6167-607X ‎Maryam Baghipur Department of Mathematics, University of Hormozgan, P. O. Box 3995, Bandar Abbas, Iran Shariefuddin Pirzada Department of Mathematics, University of Kashmir, Srinagar, India Yilun Shang Department of Computer and Information Sciences, Northumbria University, Newcastle, UK Journal Article 2020 03 02 ‎Given a simple graph $G$‎, ‎the distance signlesss Laplacian‎ ‎$D^{Q}(G)=Tr(G)+D(G)$ is the sum of vertex transmissions matrix‎ ‎$Tr(G)$ and distance matrix $D(G)$‎. ‎In this paper‎, ‎thanks to the‎ ‎symmetry of $D^{Q}(G)$‎, ‎we obtain novel sharp bounds on the distance‎ ‎signless Laplacian eigenvalues of $G$‎, ‎and in particular the‎ ‎distance signless Laplacian spectral radius‎. ‎The bounds are‎ ‎expressed through graph diameter‎, ‎vertex covering number‎, ‎edge‎ ‎covering number‎, ‎clique number‎, ‎independence number‎, ‎domination‎ ‎number as well as extremal transmission degrees‎. ‎The graphs‎ ‎achieving the corresponding bounds are delineated‎. ‎In addition‎, ‎we‎ ‎investigate the distance signless Laplacian spectrum induced by‎ ‎Indu-Bala product‎, ‎Cartesian product as well as extended double‎ ‎cover graph‎. https://toc.ui.ac.ir/article_24869_33e5843c0139fc2f041661ebaf4e1d2f.pdf