In this talk, we investigate degrees of h-polynomials of edge ideals of finite simple graphs. In particular, we provide combinatorial formulas for the degree of the $h$-polynomial for various fundamental classes of graphs such as paths, cycles, and bipartite graphs. Additionally, we characterize all connected graphs in which the sum of the Castelnuovo-Mumford regularity and the degree of the $h$-polynomial of an edge ideal. Time allowing, we will use Alexander duality to consider the $a$-invariant of cover ideals of graphs. This is joint work with Jennifer Biermann, Selvi Kara, Joseph Skelton, and Gabrial Sosa Castillo.