A Hoffman's Theorem: a revisit with new discovery

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.