Hoffman's bound for hypergraphs

Abstract

One of the best-known results in spectral graph theory is the inequality of Hoffman \[ \chi\left( G\right) \geq1-\frac{\lambda\left( G\right) }{\lambda_{\min }\left( G\right) }, \] where $\chi\left( G\right) $ is the chromatic number of a graph $G$ and $\lambda\left( G\right) ,$ $\lambda_{\min}\left( G\right) $ are the largest and the smallest eigenvalues of its adjacency matrix. In this note Hoffman's inequality is extended to weighted uniform $r$-graphs for every even $r$.