Classify connected graphs with largest normalized Laplacian eigenvalue equal to χ/(χ−1)

Determine all connected finite simple graphs G for which the largest eigenvalue λ_max(G) of the normalized Laplacian L(G)=I−D^{-1}A equals χ(G)/(χ(G)−1), where χ(G) denotes the vertex chromatic number of G.

Background

The largest eigenvalue of the normalized Laplacian L(G) satisfies the general lower bound λ_max(G) ≥ χ(G)/(χ(G)−1). The paper investigates properties of graphs for which this bound is sharp, including multiplicity and structural characterizations, and constructs families achieving equality. A complete classification remains unresolved and was explicitly posed as an open question by Sun and Das (2020).

Resolving this would unify spectral and chromatic perspectives by identifying exactly those connected graphs at the chromatic end where the normalized Laplacian’s extremal value matches the chromatic lower bound. It has implications for understanding equitable colorings with respect to D^{-1}A, multiplicities of extremal eigenvalues, and stability under graph operations such as 1-sums.