Some inequalities involving the distance signless Laplacian eigenvalues of graphs

Abstract

‎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.