On distance signless Laplacian spectrum and energy of graphs

Abdollah Alhevaz (Shahrood University of Technology)
Maryam Baghipur (Shahrood University of Technology)
Ebrahim Hashemi (Shahrood University of Technology)

Electronic Journal of Graph Theory and Applications (EJGTA)
Vol 6, No 2 (2018): Electronic Journal of Graph Theory and Applications

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.