  In 1972, A. J. Hoffman proved his celebrated theorem concerning the limit
points of spectral radii of non-negative symmetric integral matrices less than
$\sqrt{2+\sqrt{5}}$. In this paper, after giving a new version of Hoffman's
theorem, we get two generalized versions of it applicable to non-negative
symmetric matrices with fractional elements. As a corollary, we obtain another
alternative version about the limit points of spectral radii of (signless)
Laplacian matrices of graphs less than $2+ {\tiny \frac{{\;}1{\;}}{3}\left((54
- 6\sqrt{33})^{\frac{{\;}1{\;}}{3}} + (54 + 6\sqrt{33})^{\frac{{\;} 1{\;}}{3}}
\right)}$. We also discuss how our approach could be fruitfully employed to
investigate equiangular lines.
