Regularity and $a$-invariant of Cameron--Walker graphs

Abstract

Let $S$ be the polynomial ring over a field $K$ and $I\subset S$ a homogeneous ideal. Let $h(S/I,\lambda )$ be the $h$-polynomial of $S/I$ and $s=\mathrm{deg}h(S/I,\lambda )$ the degree of $h(S/I,\lambda )$. It follows that the inequality $s-r\le d-e$, where $r=\mathrm{reg}(S/I)$, $d=\dim S/I$ and $e=\mathrm{depth}S/I$, 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 $a(S/I(G))\le 0$. In the present paper, by showing the fundamental fact that every Cameron--Walker graph $G$ satisfies $a(S/I(G))=0$, a class of Cameron--Walker graphs $G$ for which $S/I(G)$ satisfies $s-r=d-e$ will be exhibited.

25 pages, 8 figures

References in corpus (1)

• Regularity and $h$-polynomials of edge ideals

Cited by in corpus (3)

• Induced matching numbers of finite graphs and edge ideals
• Homological invariants of Cameron--Walker graphs
• Matching numbers and dimension of edge ideals