On Laplacian eigenvalues of connected graphs

Let G be an undirected connected graph with n, n ⩾ 3, vertices and m edges with Laplacian eigenvalues µ1 ⩾ µ2 ⩾ ⋯ ⩾ µn−1 > µn = 0. Denote by \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu _I} = {\mu _{{r_1}}} + {\mu _{{r_2}}} + \ldots + {\mu _{{r_k}}}$$\end{document}, 1 ⩽ k ⩽ n−2, 1 ⩽ r1 < r2 < ⋯ < rk ⩽ n−1, the sum of k arbitrary Laplacian eigenvalues, with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu _{{I_1}}} = {\mu _1} + {\mu _2} + \ldots + {\mu _k}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu _{{I_n}}} = {\mu _{n - k}} + \ldots + {\mu _{n - 1}}$$\end{document}. Lower bounds of graph invariants \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu _{{I_1}}} - {\mu _{{I_n}}}$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mu _{{I_1}}}/{\mu _{{I_n}}}$$\end{document} are obtained. Some known inequalities follow as a special case.