At the end of the spectrum: Chromatic bounds for the largest eigenvalue of the normalized Laplacian

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 $\chi$.