On graded Betti numbers of edge rings of deficient complete bipartite graphs

Abstract

In this paper, we introduce deficient complete bipartite graphs and deduce a formula for graded Betti numbers of their edge rings in terms of associated combinatorial data. As a consequence, we obtain the graded Betti numbers of edge rings of complete bipartite graphs ${\mathcal{K}}_{m,n}$ and the n-crown graph ${\mathcal{C}}_{n,n}.$ Also, the projective dimension and the regularity of edge rings of these graphs are obtained.

Keywords:

• Betti numbers;edge ideals
• projective dimension
• regularity

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13D02
• 13F55
• 05C69
• 05C99

Acknowledgment

The authors are grateful to the anonymous referee for his valuable comments and helpful suggestions.