I will report on an ongoing project with Matteo Varbaro and Michela Di Marca (U Genova). In commutative algebra, Castelnuovo and Mumford introduced a notion of regularity for ideals of polynomials. In combinatorics we also use of the word "regularity", for graphs in which all vertices have the same number of neighbors. We show a surprising connection between the two notions, for arithmetically Gorenstein arrangements of projective lines. If time permits, I will discuss extensions from lines to curves, and from graphs to simplicial complexes.