Regularity and a-invariant of Cameron–Walker graphs

Abstract

Let S be a polynomial ring over a field K and I⊂S a homogeneous ideal. Let h(S/I,λ) be the h-polynomial of S/I and s=deg⁡h(S/I,λ) the degree of h(S/I,λ). It follows that the inequality s−r≤d−e, where r=reg(S/I), d=dim⁡S/I and e=depth(S/I), is satisfied and, in addition, the equality s−r=d−e holds if and only if S/I has a unique extremal Betti number. We are interested in finding a natural class of finite simple graphs G for which S/I(G), where I(G) is the edge ideal of G, satisfies s−r=d−e. Let a(S/I(G)) denote the a-invariant of S/I, i.e., a(S/I(G))=s−d. One has a(S/I(G))≤0. In the present paper, by showing the fundamental fact that every Cameron–Walker graph G satisfies a(S/I(G))=0, a classification of Cameron–Walker graphs G for which S/I(G) satisfies s−r=d−e will be exhibited.