h-Vectors of Edge Rings of odd Cycle Compositions

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