This paper describes an application of research that sits at the intersection of commutative algebra and combinatorics: Betti numbers of graphs. In particular, we describe a correspondence between simple undirected graphs and a class of ideals in a polynomial ring. We then briefly introduce some of the algebraic invariants that can be associated to the ideal and the relation of these invariants...

We initiate a classification of polynomials f : C → C of degree d having the top Betti number of the general fibre close to the maximum. We find a range in which the polynomial must have isolated singularities and another range where it may have at most one line singularity of general Morse transversal type. Our method uses deformations into particular pencils with non-isolated singularities. 2...

Recently, a universal formula for non-holomorphic modular completion of the generating functions refined BPS indices in various theories with $N=2$ supersymmetry has been suggested. It expresses through holomorphic lower ranks. Here we show that $U(N)$ Vafa-Witten theory on Hirzebruch and del Pezzo surfaces this can be used to extract themselves, thereby providing Betti numbers instanton moduli...