Almost Gorenstein edge rings of bipartite graphs and their $h$-vectors

We characterize bipartite graphs whose edge rings are almost Gorenstein. Using this characterization, we recover known results for two families of graphs. In addition, we show that if the edge ring of a bipartite graph is almost Gorenstein and its $h$-polynomial has degree at least two, then the $h$-polynomial is palindromic except possibly for the linear and next-to-leading coefficients. Moreover, we construct two bipartite graphs whose edge rings have the same $h$-vector, although only one of the rings is almost Gorenstein.