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 Tr(G)
be the diagonal matrix formed by the vertex transmissions. The distance signless Laplacian of G is defined by DQ = D(G)+Tr(G). The largest eigenvalue of DQ, written as ∂Q 1 (G), is referred to as the distance signless Laplacian spectral radius of G. In this work, we obtain several bounds on both ∂Q 1 (G) 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 ∂Q 1 (G) ≥ ∂Q 1 (Gn,δ), with equality occurring exactly when G ∼= Gn,δ. Here Gn,δ denotes the graph obtained by choosing δ vertices of Kn−1 and attaching a new vertex to them, where 1 ≤ δ ≤ n − 1. For such a graph G, we further establish that EDQ(G) ≥ EDQ(Gn,δ), and equality holds if and only if G ∼=Gn,δ. The notation EDQ(G) 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.