Least-eigenvalue minimizer for the remaining parity cases

Prove that, for every graph order n and chromatic number c satisfying 3 c c n1, the graph G(ceiling(c/2)ceil, cfloor((n-c)/2)floor) uniquely minimizes the least adjacency eigenvalue among all graphs of order n and chromatic number c, including the cases in which n and c are not both even.

Background

Fan, Yu, and Wang conjectured that the graph G(ceiling(c/2)ceil, cfloor((n-c)/2)floor) is the unique least-eigenvalue minimizer among graphs of order n and chromatic number c throughout the range 3 c c n1. The paper establishes the corresponding least-eigenvalue bound and equality characterization when both n and c are even, but does not settle the remaining parity cases.

The authors explain that when both n and c are odd, the relevant least eigenvalue satisfies a quartic equation rather than the quadratic equation available at the symmetric point. Proving the required nonnegativity inequality must then be carried out over a half-integer lattice and requires separate treatment of several endpoint cases. A unified proof for all parity cases is left unresolved.