On the Roots of σ‐Polynomials

Given a graph G of order n , the σ‐ polynomial of G is the generating function σ ( G , x ) = ∑ a i x iwhere a i is the number of partitions of the vertex set of G into i nonempty independent sets. Such polynomials arise in a natural way from chromatic polynomials. Brenti (Trans Am Math Soc 332 (1992), 729–756) proved that σ‐polynomials of graphs with chromatic number at least n − 2 had all real roots, and conjectured the same held for chromatic number n − 3 . We affirm this conjecture.