A Hoffman's Theorem: a revisit with new discovery

TL;DR

This paper revisits Hoffman's theorem, extends it to matrices with fractional elements, and explores implications for spectral radii limits of graph matrices and applications to equiangular lines.

Contribution

It provides a new version of Hoffman's theorem, generalizes it for fractional matrices, and offers alternative bounds for spectral radii of graph Laplacians.

Findings

New version of Hoffman's theorem presented

Generalizations applicable to fractional matrices

Bounds on spectral radii of graph Laplacians derived

Abstract

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.