Induced matching numbers of finite graphs and edge ideals

Abstract Let G be a finite simple graph on the vertex set V ( G ) = { x 1 , … , x n } and I ( G ) ⊂ K [ V ( G ) ] its edge ideal, where K [ V ( G ) ] is the polynomial ring in x 1 , … , x n over a field K with each deg ⁡ x i = 1 and where I ( G ) is generated by those squarefree quadratic monomials x i x j for which { x i , x j } is an edge of G. In the present paper, given integers 1 ≤ a ≤ r and s ≥ 1 , the existence of a finite connected simple graph G = G ( a , r , s ) with im ( G ) = a , reg ( R / I ( G ) ) = r and deg ⁡ h K [ V ( G ) ] / I ( G ) ( λ ) = s , where im ( G ) is the induced matching number of G and where h K [ V ( G ) ] / I ( G ) ( λ ) is the h-polynomial of K [ V ( G ) ] / I ( G ) .