  For a graph with largest normalized Laplacian eigenvalue $\lambda_N$ and
(vertex) coloring number $\chi$, it is known that $\lambda_N\geq
\chi/(\chi-1)$. Here we prove properties of graphs for which this bound is
sharp, and we study the multiplicity of $\chi/(\chi-1)$. We then describe a
family of graphs with largest eigenvalue $\chi/(\chi-1)$. We also study the
spectrum of the $1$-sum of two graphs (also known as graph joining or
coalescing), with a focus on the maximal eigenvalue. Finally, we give upper
bounds on $\lambda_N$ in terms of $\chi$.
