Weighted graphs: Eigenvalues and chromatic number

Abstract

We revisit Hoffman relation involving chromatic number $\chi$ and eigenvalues. We construct some graphs and weighted graphs such that the largest and smallest eigenvalues $\lambda$ dan $\mu$ satisfy $\lambda=(1-\chi)\mu.$ We study in particular the eigenvalues of the integer simplex $T_m^2,$ a 3-chromatic graph on $\binom {m+2}{2}$ vertices.