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.