  Let $\mathbb{K}[G]$ be the edge ring of a finite simple graph $G$.
Investigating properties of the $h$-vector of $\mathbb{K}[G]$ is of great
interest in combinatorial commutative algebra. However, there are few families
of graphs for which the $h$-vector has been explicitly determined. In this
paper, we compute the $h$-vectors of a certain family of graphs that satisfy
the odd-cycle condition, generalizing a result of the second and third named
authors. As a corollary, we obtain a characterization of the graphs in this
family whose edge rings are almost Gorenstein.
