Pseudo-Gorenstein edge rings and a new family of almost Gorenstein edge rings

In this paper, we study edge rings and their $h$-polynomials. We investigate when edge rings are pseudo-Gorenstein, which means that the leading coefficients of the $h$-polynomials of edge rings are equal to $1$. Moreover, we compute the $h$-polynomials of a special family of edge rings and show that some of them are almost Gorenstein.