On Laplacian eigenvalues of connected graphs

Abstract

Let G be an undirected connected graph with n, n ⩾ 3, vertices and m edges with Laplacian eigenvalues µ₁ ⩾ µ₂ ⩾ ⋯ ⩾ µ _n−1 > µ _n = 0. Denote by ${\mu _I} = {\mu _{{r_1}}} + {\mu _{{r_2}}} + \ldots + {\mu _{{r_k}}}$, 1 ⩽ k ⩽ n−2, 1 ⩽ r ₁ < r ₂ < ⋯ < r _k ⩽ n−1, the sum of k arbitrary Laplacian eigenvalues, with ${\mu _{{I_1}}} = {\mu _1} + {\mu _2} + \ldots + {\mu _k}$ and ${\mu _{{I_n}}} = {\mu _{n - k}} + \ldots + {\mu _{n - 1}}$. Lower bounds of graph invariants ${\mu _{{I_1}}} - {\mu _{{I_n}}}$ and ${\mu _{{I_1}}}/{\mu _{{I_n}}}$ are obtained. Some known inequalities follow as a special case.