Inequalities for Distance Signless Laplacian Matrix Under Minimum‐Degree Constraints

For a connected graph G of order n , let D ( G ) denote its distance matrix and let T r ( G ) be the diagonal matrix formed by the vertex transmissions. The distance signless Laplacian of G is defined by D Q = D ( G ) + T r ( G ). The largest eigenvalue of D Q , written as , is referred to as the distance signless Laplacian spectral radius of G . In this work, we obtain several bounds on both and on the distance signless Laplacian energy, expressed via the minimum degree δ , the Wiener index, the order, and the transmission degrees of the graph. In particular, we show that if G has minimum degree δ , then , with equality occurring exactly when G ≅ G n , δ . Here, G n , δ denotes the graph obtained by choosing δ vertices of K n −1 and attaching a new vertex to them, where 1 ≤ δ ≤ n − 1. For such a graph G , we further establish that , and equality holds if and only if G ≅ G n , δ . The notation stands for the distance signless Laplacian energy of G . We also verify that for k ‐transmission regular graphs, the distance signless Laplacian energy matches the distance energy, and we obtain a relation linking the distance signless Laplacian spectral radius with the distance spectral radius.