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

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