On distance signless Laplacian spectrum and energy of graphs - Abdollah Alhevaz; Maryam Baghipur; Ebrahim Hashemi (Electronic Journal of Graph Theory and Applications (EJGTA), 2018)
The distance signless Laplacian spectral radius of a connected graph G is the largest eigenvalue of the distance signless Laplacian matrix of G‎, ‎defined as ‎D‎Q(G) = Tr(G) + D(G)‎, ‎where D(G) is the distance matrix of G and Tr(G) is the diagonal matrix of vertex transmissions of G‎. ‎In this paper we determine some upper and lower bounds on the distance signless Laplacian spectral radius of G based on its order and independence number‎, ‎and characterize the extremal graph‎. ‎In addition‎, ‎we give an exact description of the distance signless Laplacian spectrum and the distance signless Laplacian energy of the join of regular graphs in terms of their adjacency spectrum‎.