Tales of Hoffman

Hofmman’s bound on the chromatic number of a graph states that $\chi \geq 1 - \frac {\lambda_1} {\lambda_n}$. Here we show that the same bound, or slight modifications of it, hold for several graph parameters related to the chromatic number: the vector coloring number, the $\psi$-covering number and the $\lambda$-clustering number.