Distance Signless Laplacian Eigenvalues, Diameter, and Clique Number

Let G be a connected graph of order n . Let D iag ( Tr ) be the diagonal matrix of vertex transmissions and let D ( G ) be the distance matrix of G . The distance signless Laplacian matrix of G is defined as D Q ( G ) = D iag ( Tr ) + D ( G ) and the eigenvalues of D Q ( G ) are called the distance signless Laplacian eigenvalues of G . Let ∂ Q 1 ( G ) ≥ ∂ Q 2 ( G ) ≥ · · · ≥ ∂ Q n ( G ) be the distance signless Laplacian eigenvalues of G . The largest eigenvalue ∂ Q 1 ( G ) is called the distance signless Laplacian spectral radius. We obtain a lower bound for ∂ Q 1 ( G ) in terms of the diameter and order of G . With a given interval I , denote by m D Q ( G ) I the number of distance signless Laplacian eigenvalues of G which lie in I . For a given interval I , we also obtain several bounds on m D Q ( G ) I in terms of various structural parameters of the graph G , including diameter and clique number.