Spectral bounds for the vulnerability parameters of graphs

The scattering number, integrity, and tenacity are preferred for evaluating the vulnerability and reliability of a communication network. In this paper, we establish a tight signless Laplacian spectral radius condition to guarantee the scattering number of a connected graph G with the minimum degree delta\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} to be at most zero. As well as new tight bounds for the integrity and tenacity of graphs in terms of their (normalized) Laplacian eigenvalues are also included. Our results extend the corresponding results on regular graphs due to Li et al. (Future Gener Comput Syst 83:450-453, 2018), respectively.