Laplacian Controllability for Graphs with Integral Laplacian Spectrum

If G is a graph with n vertices, LG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_G$$\end{document} is its Laplacian matrix, and b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {b}$$\end{document} is a binary vector of length n, then the pair (LG,b)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(L_G, \mathbf {b})$$\end{document} is said to be controllable, and we also say that G is Laplacian controllable for b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {b}$$\end{document}, if b\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {b}$$\end{document} is non-orthogonal to any of the eigenvectors of LG\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L_G$$\end{document}. It is known that if G is Laplacian controllable, then it has no repeated Laplacian eigenvalues. If G has no repeated Laplacian eigenvalues and each of them is an integer, then G is decomposable into a (dominate) induced subgraph, say H, and another induced subgraph with at most three vertices. We express the Laplacian controllability of G in terms of that of H. In this way, we address the question on the Laplacian controllability of cographs and, in particular, threshold graphs.