Let G be a connected graph with n vertices, m edges and having distance signless Laplacian eigenvalues ρ₁≥ ρ₂ ≥ … ≥ ρ_n≥ 0. For any real number α ≠ 0, let \({m_\alpha }\left( G \right) = \sum\nolimits_{i = 1}^n {\rho _i^\alpha } \) be the sum of α^th powers of the distance signless Laplacian eigenvalues of the graph G. In this paper, we obtain various bounds for the graph invariant m_α(G), which connects it with different parameters associated to the structure of the graph G. We also obtain various bounds for the quantity DEL(G), the distance signless Laplacian-energy-like invariant of the graph G. These bounds improve some previously known bounds. We also pose some extremal problems about DEL(G).

让ģ是具有连通图Ñ顶点，米缘并具有距离无符号Laplace特征值ρ ₁ ≥ρ ₂ ≥ ...≥ρ _Ñ ≥ 0。对于任何实数α ≠0，让\（{M_ \阿尔法} \左（ G \ right）= \ sum \ nolimits_ {i = 1} ^ n {\ rho _i ^ \ alpha} \）是图G的距离无符号拉普拉斯特征值的α^次方和。在本文中，我们获得用于图形不变各个边界米_α（g ^），并将其与与图G的结构相关联的不同参数联系起来。我们还获得了数量DEL（G）的各种界线，即图G的距离无符号拉普拉斯能量式不变量。这些界限改善了一些先前已知的界限。我们还提出了一些有关DEL（G）的极端问题。