The $h$-vectors of the edge rings of a special family of graphs

The $h$-vectors of homogeneous rings are one of the most important invariants that often reflect ring-theoretic properties. On the other hand, there are few examples of edge rings of graphs whose $h$-vectors are explicitly computed. In this paper, we compute the $h$-vector of a special family of graphs, by using the technique of initial ideals and the associated simplicial complex.