On the Distance Signless Laplacian Spectrum of Graphs

Abstract

Let G be a simple, connected graph, D(G) be the distance matrix of G, and Tr(G) be the diagonal matrix of vertex transmissions of G. The distance signless Laplacian matrix of G is defined as $$D^Q(G)=Tr(G)+D(G)$$ , and the distance signless Laplacian spectral radius of G is the largest eigenvalue of $$D^{Q}(G)$$ . In this paper, we study Nordhaus–Gaddum-type inequalities for distance signless Laplacian eigenvalues of graphs and present some new upper and lower bounds on the distance signless Laplacian spectral radius of G and of its line graph L(G), based on other graph-theoretic parameters, and characterize the extremal graphs. Further, we study the distance signless Laplacian spectrum of some graphs obtained by operations.