Koszul graded Möbius algebras and strongly chordal graphs

The graded M & ouml;bius algebra of a matroid is a commutative graded algebra which encodes the combinatorics of the lattice of flats of the matroid. As a special subalgebra of the augmented Chow ring of the matroid, it plays an important role in the recent proof of the Dowling-Wilson Top Heavy Conjecture. Recently, Mastroeni and McCullough proved that the Chow ring and the augmented Chow ring of a matroid are Koszul. We study when graded M & ouml;bius algebras are Koszul. We characterize the Koszul graded M & ouml;bius algebras of cycle matroids of graphs in terms of properties of the graphs. Our results yield a new characterization of strongly chordal graphs via edge orderings.